with an enormous amount of visual data online , web and social media are among the most important sources of data for vision research .vision datasets such as imagenet  , pascal voc  and ms coco  have been created from google or flickr by harnessing human intelligence to filter out the noisy images and label object locations .the resulting clean data has helped significantly advance performance on relevant tasks  .for example , training a neural network on imagenet followed by fine - tuning on pascal voc has led to the state - of - the - art performance on the object detection challenge  .but human supervision comes with a cost and its own problems ( inconsistency , incompleteness and bias  ) .therefore , an alternative , and more appealing way is to learn visual representations and object detectors from the web data directly , without using any manual labeling of bounding boxes .but the big question is , can we actually use millions of images online without using any human supervision ?  ]  in fact , researchers have pushed hard to realize this dream of learning visual representations and object detectors from web data .these efforts have looked at different aspects of webly supervised learning such as :  * * what are the good sources of data ?* researchers have tried various search engines ranging from text / image search engines  to flickr images  . * * what types of data can be exploited ?* researchers have tried to explore different types of data , like images - only  , images - with - text  or even images - with--grams  ) .* * how do we exploit the data ? * extensive algorithms ( probabilistic models  , exemplar based models  , deformable part models  , self organizing map  ) have been developed .* * what should we learn from web data ?* there has been lot of effort ranging from just cleaning data  to training visual models  , to even discovering common - sense relationships  .nevertheless , while many of these systems have seen orders of magnitudes larger number of images , their performance has never matched up against contemporary methods that receive extensive supervision from humans .why is that ?  of course the biggest issue is the data itself : 1 ) it contains noise , and 2 ) is has bias - image search engines like google usually operate in the high - precision low - recall regime and tend to be biased toward images where a single object is centered with a clean background and a canonical viewpoint  . but is it just the data ?we argue that it is not just the data itself , but also the ability of algorithms to learn from large data sources and generalize .for example , traditional approaches which use hand - crafted features ( hog  ) and classifiers like support vector machines  have very few parameters ( less capacity to memorize ) and are therefore unlikely to effectively use large - scale training data . on the other hand ,memory based nearest neighbors classifiers can better capture the distribution given a sufficient amount of data , but are less robust to the noise .fortunately , convolutional neural networks ( cnns )  have resurfaced as a powerful tool for learning from large - scale data : when trained with imagenet  ( m images ) , it is not only able to achieve state - of - the - art performance for the same image classification task , but the learned representation can be readily applied to other relevant tasks  . attracted by their amazing capability to harness large - scale data , in this paper , we investigate webly supervised learning for cnns ( see figure [ fig : teaser ] ) .specifically , 1 ) we present a two - stage webly supervised approach to learning cnns .first we show that cnns can be readily trained for easy categories using images retrieved by search engines .we then adapt this network to hard ( flickr style ) web images using the relationships discovered in easy images .2 ) we show webly supervised cnns also generalize well to relevant vision tasks , giving state - of - the - art performance compared to imagenet pretrained cnns if there is enough data .3 ) we show state - of - the - art performance on voc data for the scenario where not a single voc training image is used - just the images from the web .4 ) we also show competitive results on scene classification .we believe this paper opens up avenues for exploitation of web data to achieve next cycle of performance gain in vision tasks ( and at no human labeling costs ! ) .   driven by cnns , the field of object detection has seen a dramatic churning in the past two years , which has resulted in a significant improvement in the state - of - the - art performance .but as we move forward , how do we further improve performance of cnn - based approaches ?we believe there are two directions .the first and already explored area is designing deeper networks  .we believe a more promising direction is to feed more data into these networks ( in fact , deeper networks would often need more data to train ) .but more data needs more human labeling efforts .but data labeling in terms of bounding boxes can be very cumbersome and expensive .therefore , if we can exploit web data for training cnns , it would help us move from million to billion image datasets in the future . in this paper, we take the first step in demonstrating : 1 ) cnns can be trained effectively by just exploiting web data at much larger scales ; 2 ) competitive object detection results can be obtained without using a single bounding box labels from humans .mining high - quality visual data and learning good visual representation for recognition from the web naturally form two aspects of a typical chicken - and - egg problem in vision .on one hand , clean and representative seed images can help build better and more powerful models ; but on the other hand , models that recognize concepts well are crucial for indexing and retrieving image sets that contain the concept of interest . how to attack this problem has long been attractive to both industry and academia .* from models to data * : image retrieval  is a classical problem in this setting .it is not only an active research topic , but also fascinating to commercial image search engines and photo - sharing websites since they would like to better capture data streams on the internet and thus better serve user s information need . over the years ,various techniques ( click - through data ) have been integrated to improve search engine results .note that , using pretrained models ( cnn  ) to clean up web data also falls into this category , since extensive human supervision has already been used .  * from data to models * : a more interesting and challenging direction is the opposite - can models automatically discover the hidden structures in the data and be trained directly from web data ?many people have pushed hard in this line of research .for example , earlier work focused on jointly modeling images and text and used text based search engines for gathering the data  .this tends to offer less biased training pairs , but unfortunately such an association is often too weak and hard to capture , since visual knowledge is usually regarded as common sense knowledge and too obvious to be mentioned in the text  .as the image search engines became mature , recent work focused on using them to filter out the noise when learning visual models  . but using image search engines added more bias to the gathered data  . to combat both noise and data bias ,recent approaches have taken a more semi - supervised approach .in particular , proposed iterative approaches to jointly learn models and find clean examples , hoping that simple examples learned first can help the model learn harder , more complex examples  .however , to the best of our knowledge , human supervision is still a clear winner in performance , regardless of orders of magnitudes more data seen by many of these web learners .our work is also closely related to another trend in computer vision : learning and exploiting visual representation via cnns  .however , learning these cnns from noisy labeled data  is still an open challenge .following the recent success of convolutional networks and curriculum learning  , we demonstrate that , while directly training cnns with high - level or fine - grained queries ( random proper nouns , abstract concepts ) and noisy labels ( flickr tags ) can still be challenging , a more learning approach might provide us the right solution . specifically , one can bootstrap cnn training with easy examples first , followed by a more extensive and comprehensive learning procedure with similarity constraints to learn visual representations .we demonstrate that visual representations learned by our algorithm performs very competitively as compared to imagenet trained cnns .  finally , our paper is also related to learning from weak or noisy labels  .there are some recent works showcasing that cnns trained in a weakly supervised setting can also develop detailed information about the object intrinsically  .however , different from the assumptions in most weakly - supervised approaches , here our model is deprived of clean human supervision altogether ( instead of only removing the location or segmentation ) .most recently , novel loss layers have also been introduced in cnns to deal with noisy labels  . on the other hand ,we assume a vanilla cnn is robust to noise when trained with simple examples , from which a relationship graph can be learned , and this relationship graph provides powerful constraints when the network is faced with more challenging and noisier data .our goal is to learn deep representations directly from the massive amount of data online .while it seems that cnns are designed for big data - small datasets plus millions of parameters can easily lead to over - fitting , we found it is still hard to train a cnn naively with random image - text / tag pairs .for example , most flickr tags correspond to meta information and specific locations , which usually results in extremely high intra - tag variation .one possibility is to use commercial text - based image search engine to increase diversity in the training data .but if thousands of query strings are used some of them might not correspond to a visualizable concept and some of the query strings might be too fine grained ( random names of a person or abstract concepts ) . these non - visualizable concepts and fine - grained categoriesincur unexpected noise during the training process7000 noun phrases randomly sampled from the web ( m images ) , but it does not converge . ] .one can use specifically designed techniques  and loss layers  to alleviate some of these problems .but these approaches are based on estimating the empirical noise distribution which is non - trivial .learning the noise distribution is non - trivial since it is heavily dependent on the representation , and weak features ( hog or when the network is being trained from scratch ) often lead to incorrect estimates . on the other hand , for many basic categories commonly used in the vision community , the top results returned by google image search are pretty clean .in fact , they are so clean that they are biased towards iconic images where a single object is centered with a clean background in a canonical viewpoint  .this is good news for learning algorithm to quickly grasp the appearance of a certain concept , but a representation learned from such data is likely biased and less generalizable .so , what we want is an approach that can learn visual representation from flickr - like images .  inspired by the philosophy of curriculum learning  , we take a two - step approach to train cnns from the web . in curriculum learning ,the model is designed to learn the easy examples first , and gradually adapt itself to harder examples . in a similar manner , we first train our cnn model from scratch using easy images downloaded from google image search .once we have this representation learned we try to feed harder flickr images for training .note that training with flickr images is still difficult because of noise in the labels .therefore , we apply constraints during fine - tuning with flickr images . these constraints are based on similarity relationships across different categories .specifically , we propose to learn a relationship graph and initial visual representation from the easy examples first , and later during fine - tuning , the error can back - propagate through the graph and get properly regularized .the outline of our approach is shown in figure [ fig : outline ] .   as noted above , common categories used in vision nowadaysare well - studied and search engines give relatively clean results .therefore , instead of using random noun phrases , we obtained three lists of categories from imagenet challenge  , sun database  and neil knowledge base  .imagenet syn - sets are transformed to its surface forms by just taking the first explanation , with most of them focusing on object categories .to better assist querying and reducing noise , we remove the suffix ( usually correspond to attributes , indoor / outdoor ) of the sun categories . since neil is designed to query search engines , its list is comprehensive and favorable , we collected the list for objects and attributes and removed the duplicate queries with imagenet .the category names are directly used to query google for images . apart from removing unreadable images ,no pre - processing is performed .this leave us with images for each query .all the images are then fed directly into the cnn as training data .  for fair comparison, we use the same architecture ( besides the output layer ) as the blvc reference network  , which is a slight variant of of the original network proposed by  .the architecture has five convolutional layers followed by two fully connected layers .after seventh layer , another fully connected layer is used to predict class labels .   after converging, the initial network has already learned favorable low - level filters to represent the `` visual world '' outlined by google image search .however , as mentioned before , this `` visual world '' is biased toward clean and simple images .for example , it was found that more than 40% of the cars returned by google are viewed from a 45 degree angle  .moreover , when a concept is a product , lots of the images are wallpapers and advertisements with artificial background , with the product centered and pictured from the best selling view . on the other hand , photo - sharing websites like flickr have more realistic images since the users upload their own photos .though photographic bias still exists , most of the images are closer - looking to the visual world humans experience everyday .datasets constructed from them are shown to generalize better  . therefore , as a next step , we aim to narrow the gap by fine - tuning our representation on flickr images .  for fine - tuning the network with hard flickr images, we again feed these images as - is for training , with the tags as class labels . while we are getting more realistic images, we did notice that the data becomes noisier .powerful as cnns , they are still likely to be diluted by the noisy examples over the fine - tuning process1500 categories and close - to - uniform label distribution , a cnn converged on google images yields an entropy .8 , whereas flickr gives .0 . note that complete random noise will give (1500)=7.3 and perfectly separable signal close to 0.0 . ] .in an noisy open - domain environment , mistakes are unavoidable . but humans are more intelligent : we not just learn to recognize concepts independently , but also build up interconnections and develop theories to help better understand the world  .inspired by this , we want to train cnns with such relationships - with their simplest form being pair - wise look - alike ones  . such a relationship graph can provide more information of the class and regularize / constrain the network training .a motivating example is `` iphone '' .while google mostly returns images of the product , on flickr it is often used to specify the device a photo is taken with - as a result , virtually any image can be tagged as `` iphone '' . knowing similar - looking categories to `` iphone ''can intuitively help here .one way to obtain relationships is through extra knowledge sources like wordnet  .however , they are not necessarily developed for the visual domain .instead , we take a data - driven approach to discover relationships in our data : we assume the network will intrinsically develop connections between different categories when clean examples are offered , and all we have to do is to distill the knowledge out .  we take a simple approach by just testing our network on the training set , and take the confusion matrix as the relationships .mathematically , for any pair of concepts and , the relationship is defined as : where is the set of indexes for images that belong to concept , is the cardinality function , and given pixel values , is the network s belief on how likely image belongs to concept .we want our graph to be sparse , therefore we just used the top ( in our experiments ) and re - normalized the probability mass .after constructing the relationship graph , we put this graph ( represented as a matrix ) on top of the seventh layer of the network , so that now the soft - max loss function becomes : where is the class label . in this way, the network is trained to predict the context of a category ( in terms of relationships to other categories ) , and the error is back - propagated through the relationship graph to lower layers .note that , this extra layer is similar to  , in which is used to characterize the label - flip noise .different from them , we do not assume all the categories are _ mutually exclusive _ , but instead _inter related_. for example , `` cat '' is a hyper - class of `` siamese cat '' , and it is reasonable if the model believes some examples of `` siamese cat '' are more close to the average image of a `` cat '' .please see section [ sec : exp ] for our empirical validation of this assumption . for fear of semantic drift , in this paperwe keep the initially learned graph structure fixed , but it would be interesting to see how updating the relationship graph performs ( like  ) .features of the seed images are fired on edgebox proposals ( purple boxes ) from other images for nearest neighbors ( red boxes ) , which are then merged to form subcategories .noisy subcategories are purged with density estimation  . [fig : loc ] ]   until now , we have focused on learning a webly - supervised cnn representation based on classification loss . in order to train a webly - supervised object detectorwe still need to clean the web data and localize the objects in those images to train a detector like r - cnn  .note that this is a non - trivial task , since : 1 ) the cnn is only trained to distinguish a closed set of classes , unnecessarily aware of all the negative visual world , background clutter ; 2 ) the classification loss encourages the representation to be spatially _ invariant _ ( , the network should output `` orange '' regardless of where it exists in the image or how many there are ) , which can be a serious issue for localization .we now describe our subcategory discovery based approach similar to  to clean data and localize objects .the whole process is illustrated in figure [ fig : loc ] .  * seeds * : we use the full images returned by google as seed bounding boxes .this is based on google s bias toward images with a single centered object and a clean background .  * nearest neighbor propagation * : for each seed , we train an exemplar - lda  detector using our trained features .negative statistics for e - lda are computed over all the downloaded images .this e - lda detector is then fired on the remaining images to find its top nearest neighbors . for efficiency , instead of checking all possible windows on each image , we use edgebox  to propose candidate ones , which also reduces background noise .we set =10 in our experiments .  *clustering into subcategories * : we then use a publicly - available variant of agglomerative clustering  where the nearest neighbor sets are merged iteratively from bottom up to form the final subcategories based on e - lda similarity scores and density estimation .note that this is different from  , but gives similar results while being much more efficient .some example subcategories are shown in figure [ fig : subcat ] .finally , we train a r - cnn  detector for each category based on all the clustered bounding boxes .random patches from yfcc  are used as negatives .the naive approach would be using the positive examples as - is .typically , hundreds of instances per category are available for training .while this number is comparable to the voc 2007 trainval set  , we also tried to increase positive bounding boxes using two strategies :  * edgebox augmentation ( ea ) * : we follow  to augment the positive training examples .we again use edgebox  to propose regions of interest on images . whenever a proposal has a .5 overlapping ( measured by intersection over union ) with any of the positive bounding box , we add it for training .* category expansion ( ce ) * : one big advantage of internet is its nearly infinite data limit .here we again use the relationship graph to look for similar categories for more training examples . after verification the semantic - relatedness with wordnet  , we add the examples into training dataset .we believe the extra examples should allow better generalization .note both these strategies are only used to increase the amount of positive data for the final svm to be trained in r - cnn .we do not re - train our cnn representations using these strategies .we now describe our experimental results .our goal is to demonstrate that the visual representation learned using two - step webly supervised learning is meaningful . for this, we will do four experiments : 1 ) first , we will show that our learned cnn can be used for object detection . here , we use the approach similar to r - cnn  where we will fine - tune our learned cnn using voc data .this is followed by learning svm - detectors using cnn features .2 ) we will also show that our cnn can be used to clean up the web data : that is , discover subcategories and localize the objects in web images .3 ) we will train detectors using the cleaned up web data and evaluate them on voc data .note in this case , we will not use any voc training images .we will only use web images to train both the cnn and the subsequent svms .4 ) finally , we will show scene classification results to further showcase the usefulness of the trained representation .all the networks are trained with the caffe toolbox  . in totalwe have 2,240 objects , 89 attributes , and 874 scenes .two networks are trained on google : 1 ) the object - attribute network ( googleo ) , where the output dimension is 2,329 , and 2 ) all included network ( googlea ) , where the output dimension is 3,203 . for the first network, .5 million images are downloaded from google image search . combining scene images ,.1 million images are used in the second network .we set the batch size as 256 and start with a learning rate of 0.01 . with xavier initialization  , the learning rate is reduced by a factor of 10 after every 150k iterations , and we stop training at 450k iterations . for two - stage training , googleo is then fine - tuned with .2 million flickr images .we tested both with ( flickrg ) and without ( flickrf ) the relationship graph as regularization .fine - tuning is performed for a total of 100k iterations , with a step size of 30k . as baseline, we also report numbers for cnn learned using flickr images alone ( flickrs ) and combined google+flickr images ( gfall ) .note in case of gfall , neither two stage learning or relationship graph constraint is used .* is confusion matrix informative for relationships ? * we first want to show if the network has learned to discover the look - alike relationships between concepts in the confusion matrix . to verify the quality of the network ,we take the googleo net and visualize the top-5 most confusing concepts ( including self ) to some of the categories . to ensure our selectionhas a good coverage , we first rank the diagonal of the confusing matrix ( accuracy ) in the descending order .then we randomly sample 3 categories from the top-100 , bottom-100 , and middle-100 from the list .the visualization and explanations can be found in figure [ fig : rel ] .we can see that the top relationships learned are indeed reasonable .  * *     next , we test our webly trained cnn model for object detection on the pascal voc . following the r - cnn pipeline , two sets of experimentsare performed on voc 2007 .first , we directly test the generalizability of cnn - representations learned without fine - tuning on voc data .second , we fine tune the cnn by back - propagating the error end - to - end using pascal trainval set .the fine - tuning procedure is performed 100k iteration , with a step size of 20k . in both cases , features are extracted to represent patches , and a svm is learned to produce the final score .we report numbers for all the cnns on voc 2007 data in table [ tab : voc_2007 ] .several interesting notes :  * despite the search engine bias and the noise in the data , our two - stage cnn with graph regularization is on par with imagenet - trained cnn .* training a network directly on noisy and hard flickr images hurt the learning process .for example , flickrs gives the worst performance and in fact when a cnn is trained using all the images from google and flickr it gives a map of 40.5 , which is substantially lower than our map .* the proposed two - stage training strategy effectively takes advantage of the more realistic data flickr provides . without graph regularizationwe achieve a map of 43.4 ( flickrf ) .however , adding the graph regularization brings our final flickrg network on par with imagenet ( map = 44.7 ) .we use the same cnns for voc 2012 and report results in table [ tab : voc_2012 ] . in this case , our networks outperform the imagenet pretrained network even after fine - tuning ( 200k iterations , 40k step size ) . note that the original r - cnn paper fine - tuned the imagenet cnn using train data alone and therefore reports lower performance  . for fairness , we fine - tuned both imagenet network and our networks on combined trainval images ( imagenet - tv ) . in both voc 2007 and 2012 ,our webly supervised cnns tend to work better for vehicles , probably because we have lots of data for cars and other vehicles ( classes ) . on the other hand, imagenet cnn seems to outperform our network on animals  ( cat ) .this is probably because imagenet has a lot more data for animals .it also suggests our cnns can potentially benefit from more animal categories .* does web supervision work because the image search engine is cnn - based ?* one possible hypothesis can be that our approach performs comparably to imagenet - cnn because google image search itself uses a trained cnn . to testif this hypothesis is true , we trained a separate cnn using neil images downloaded from google before march 2013 ( pre - cnn based image search era ) . despite the data being noisier and less ( per category ) , we observe % performance fall compared to a cnn trained with november 2014 data on the same categories .this indicates that the underlying cnn in google image search has minimal effect on the training procedure and our network is quite robust to noise .   in this subsection , we are interested to see if we can detect objects without using a single pascal training image .we believe this is possible since we can localize objects automatically in web images with our proposed approach ( see section [ sec : loc ] ) .please refer to figure [ fig : subcat ] for the qualitative results on the training localization we can get with features .compared to  , the subcategories we get are less homogeneous ( people are not well - aligned , objects in different view points are clustered together ) . but just because of this more powerful representation ( and thus better distance metric ) , we are able to dig out more signal from the training set - since semantically related images can form clusters and wo nt be purged as noise when an image is evaluated by its nearest neighbors .  using localized objects , we train r - cnn based detectors to detect objects on the voc 2007 test set .we compare our results against  , who used google -grams to expand the categories ( `` horse '' is expanded to `` jumping horse '' , `` racing horse '' ) and the models were also directly trained from the web .the results are shown in table [ tab : uns ] . for our approach , we try five different settings : 1 ) googleo : features are based on googleo cnn and the bounding boxes are also extracted only on easy google images ; 2 ) googlea : using googleo to extract features instead ; 3 ) flickrg : features are based on flickrg instead ; 4 ) flickrg - ea : the same flickr features are used but with edgebox augmentation ; 5 ) flickrg - ce : the flickr features are used but the positive data includes examples from both original and expanded categories . from the results , we can see that in all cases the cnn based detector boosts the performance a lot .this demonstrates that our framework could be a powerful way to learn detectors for arbitrary object categories without labeling any training images .we plan to release a service for everyone to train r - cnn detectors on the fly .the code will also be released .     in this section, we would like to gain more insights about the potential issues of our webly supervised object detection pipeline .we took the results from our best model ( flickr - c ) and fed them to the publicly available diagnosis tool  .figure [ fig : diag ] and [ fig : fp ] highlight some of the interesting observations we found .firstly , localization error accounts for a majority of the false positives . since google image search do not provide precise location information , the background is inevitably included when the detector is trained ( aeroplane , dining table ) .multiple instances of an object can also occur in the image , but the algorithm has no clue that they should be treated as separate pieces ( bottle ) .moreover , since our cnn is directly trained on full images , the objective function also biases the representation to be invariant ( to spatial locations , ) .all these factors caused localization issues .second , we did observe some interesting semantic drift between pascal categories and google categories .for example , bicycle can also mean motorcycle on google .sense disambiguation for this polysemous word  is needed here .also note that our person detector is confused with cars , we suspect it is because `` caprice '' was added as a related category but it can also mean a car ( `` chevy caprice '' ) . how to handle such issues is a future research topic .   to further demonstrate the usage of cnn features directly learned from the web , we also conducted scene classification experiments on the mit indoor-67 dataset  .for each image , we simply computed the feature vector , which has 4096 dimensions .we did not use any data augmentation or spatial pooling technique , with the only pre - processing step being normalizing the feature vector to unit length  .the default svm parameters ( =1 ) were fixed throughout the experiments .table [ tab : mit67 ] summarizes the results on the default train / test split .we can see our web based cnns achieved very competitive performances : all the three networks achieved an accuracy at least on par with imagenet pretrained models .fine - tuning on hard images enhanced the features , but adding scene - related categories gave a huge boost to 66.5 ( comparable to the cnn trained on places database  , 68.2 ) .this indicates cnn features learned directly from the web are generic and quite powerful .moreover , since we can easily get images for semantic labels ( actions , -grams , ) other than objects or scenes from the web , webly supervised cnn bears a great potential to perform well on many relevant tasks - with the cost as low as providing a query list for that domain .we have presented a two - stage approach to train cnns using noisy web data .first , we train cnn with easy images downloaded from google image search .this network is then used to discover structure in the data in terms of similarity relationships .then we fine - tune the original network on more realistic flickr images with the relationship graph .we show that our two - stage cnn comes close to the imagenet pretrained - cnn on voc 2007 , and outperforms on voc 2012 .we report state - of - the - art performance on voc 2007 without using any voc training image .finally , we will like to differentiate webly supervised and unsupervised learning .webly supervised learning is suited for semantic tasks such as detection , classification ( since supervision comes from text ) . on the other hand ,unsupervised learning is useful for generic tasks which might not require semantic invariance ( , 3d understanding , grasping ) .this research is supported by onr muri n000141010934 , yahoo - cmu inmind program and a gift from google .ag and xc were partially supported by bosch young faculty fellowship and yahoo fellowship respectively .the authors would also like to thank yahoo ! for a computing cluster and nvidia for tesla gpus .

Abstract: we present an approach to utilize large amounts of web data for learning cnns . specifically inspired by curriculum learning , we present a two - step approach for cnn training . first , we use easy images to train an initial visual representation . we then use this initial cnn and adapt it to harder , more realistic images by leveraging the structure of data and categories . we demonstrate that our two - stage cnn outperforms a fine - tuned cnn trained on imagenet on pascal voc 2012 . we also demonstrate the strength of webly supervised learning by localizing objects in web images and training a r - cnn style  detector . it achieves the best performance on voc 2007 where no voc training data is used . finally , we show our approach is quite robust to noise and performs comparably even when we use image search results from march 2013 ( pre - cnn image search era ) .


after their introduction over a quarter - century ago , quantum weak values have consistently found themselves at the center of controversy . indeed , the original paper details how one can postselect a weak ( i.e. , noisy ) measurement of a spin- operator for an electron ( using a sequence of two stern - gerlach apparatuses ) to obtain a _ conditioned expectation value _ that approximates a weak value with an anomalously large value of .the question whether this strange average value has any physical meaning pertaining to the spin has since plagued the concept of the weak value ( e.g. , ) .the most recent addition to this controversy considers a superficially similar example consisting of a classical coin that has its two faces noisily measured , then _ disturbed _ , and finally conditioned to produce an anomalous average value of heads .the conclusion drawn from their study ( which has been heavily criticized ) is that strange weak values may be understood entirely as classical disturbance effects , making them not `` quantum . '' in fact , every element of this simple example of how intermediate disturbance can cause strange postselected averages of noisy signals has been previously demonstrated , and corroborates our published work : not only did we emphasize a similar disturbance example using a colored marble in our systematic investigation of generalized observable measurements , but we also carefully highlighted the potential role of invasive measurements in studies linking strange conditioned averages ( including weak values ) to violations of generalized leggett - garg inequalities ( which were designed to test for `` quantum '' behavior in macroscropic systems ) .it is now well - established that any hidden - variable model that can produce strange conditioned averages like the weak value must include some form of intermediate disturbance ( see also ) .the more interesting question to raise is not whether a particular strange conditioned average may be explained as classical disturbance , but rather whether such models of disturbance can also reproduce the complete behavior of the weak value as its physical parameters are varied .  in this paper, we revisit this question in order to dispel the abundant confusion about weak values still present in the literature , and emphasize that a strange weak value is nonclassical in precisely the same manner that a single quantum particle can be considered to be nonclassical . specifically, strange weak values fundamentally arise from _ interference _( i.e. , superoscillations ) , and thus also appear in any wave - like field theory , such as classical optics , in a straightforward way . in such a classical field theory ,anomalous weak values do faithfully indicate physical wave properties , despite how counter - intuitive their predictions may seem .for example , the orbital part of the poynting vector field of optical vortex beams , or evanescent fields , can show anomalous local momentum distributions that are precisely equal to strange weak values .therefore , as with any quantum interference effect , only the fact that discrete and independent random events can be measured ( as opposed to attenuated wave intensities ) will distinguish whether the statistics producing a strange weak value are truly quantum mechanical in origin .( alternatively , entangling the degrees of freedom of distinct particles will not have a simple classical field interpretation , e.g. , . ) nevertheless , even in the case of discrete measurement events the large number of measurements needed to statistically resolve such a weak value still imply that it is best considered as a dynamical physical variable for the effective ( classical ) mean field , and not necessarily to each individual quantum particle _ a priori _ . to emphasize these subtle points, we carefully review several complementary approaches to deriving and understanding the weak value , paying special attention to its role as an ideal estimate , an experimentally measurable conditioned average , and as a classical dynamical variable for reduced quantum state evolution .this detailed treatment aims to supplement the simplified introduction to the experimental applications of weak values in with an expanded theoretical discussion that highlights their pervasive and under - appreciated role throughout the quantum formalism .  in what followswe also emphasize the often overlooked connection between weak values and joint quasiprobability distributions ( such as the kirkwood - dirac , terletsky - margenau - hill , and the various moyal phase space distributions ) that determine conditioned observable estimates .notably , to obtain a strange weak value outside the usual eigenvalue bounds , these joint quasiprobability distributions must become negative as a consequence of the nonclassical quantum interference between probability amplitudes .the best known examples of this intrinsic negativity from quantum interference occur in the moyal phase space distributions , such as the wigner distribution for quadratures , or the glauber - sudarshan distribution for coherent state amplitudes , which show such negativity with nonclassical optical states .indeed , for single particles such negativity in quasiprobability distributions has been proven to be an equivalent notion of `` nonclassicality '' as the need for _ contextual _ hidden variables ( in the sense of bell - kochen - specker ) , and formally arises from the usual operator non - commutativity of quantum mechanics .this connection between contextuality and strange weak values was emphasized in a recent proof by pusey , as well as an earlier study by tollaksen , and is consistent with the established understanding that strange weak values arise fundamentally from ( quantum ) interference .it follows that if a classical model as in could really mimic the detailed functional structure of the weak value , then it would also be able to simulate other features that are normally considered to be quantum mechanical .we further emphasize this latter point by reviewing the deep connections between weak values and the classical dynamical variables of the hamilton - jacobi formalism , both in its fully quantum generalization that is equivalent to the schrdinger equation , and in the resulting classical limit .this paper is organized as follows . in section [ sec :estimates ] we derive and discuss how the weak value is the best statistical estimate for the average ( but unmeasured ) observable value for the times between two known measurement events . in section[ sec : measurement ] we discuss three approaches for experimentally verifying the weak value as the appropriate such estimate : weak von neumann coupling , weak generalized observable measurements , and as the physical dynamical variables for reduced state evolution . in section[ sec : quasiprob ] we explicitly connect the weak value to quasiprobabilities , focusing on the terletsky - margenau - hill , kirkwood - dirac , and wigner distributions , and explicitly connect weak values to classical mean - field dynamical variables using the hamilton - jacobi quantum - classical correspondence .we conclude in section [ sec : conclusion ] .estimates conditioned on two boundaries that bracket the time interval  ] , the weak value ( thin , green ) coincides with the expectation value ( dot - dashed , black ) when the postselection is consistent with the natural oscillation .otherwise , ( dashed , black ) displays a jump at time , while ( red ) smoothly connects the boundaries , still passing through the same points of certain ( blue dots ) .the shaded regions exceed the eigenvalue bounds of , indicating the inconsistency between the natural evolution and the observed boundaries . ]most of the controversy surrounding weak values rests upon their common ( but unnecessary ) association with an alternative time - symmetric approach to the quantum theory that involves two state vectors . in this time - symmetric approach , one forward - propagates a state - vector from an initial time to in the usual way; however , one also _back - propagates _ a second state - vector from a final time to .while the initial state vector corresponds to a preparation procedure , the final state vector corresponds to a _procedure .interestingly , the best estimate of the average ( unmeasured ) value for an observable at any time in the interval  ] .the question raised in ref . is whether such a classical model with noisy estimation and disturbance is sufficient to explain eq . .as we will show in what follows , such a classical disturbance explanation is difficult to defend when the many roles of this weak value expression are examined in detail .   for completenesswe now show how this weak value formula can be obtained as the optimal estimate that minimizes a statistical uncertainty metric , justifying its interpretation as a best estimate .suppose one wishes to estimate the best average values for an observable given a initial preparation , followed by a projective measurement in a particular basis that does not commute with .for each , we can guess a value that estimates the average of given that the specific result was observed .this procedure formally constructs a hermitian observable that is measured by the procedure , which contains the estimates for every .the goal is then to determine the optimal such estimates of conditioned on each measured result .  to accomplish this goal, we must define a measure for how close the estimate is to the target observable .a natural choice for such a measure is the weighted trace distance between two observables  which can generally depend upon any positive prior bias .we can interpret this measure as specifying the shortest _ geometric distance _ in operator space , weighted by the _ statistical prior _ bias of .since and need not commute , there is generally no straightforward operational interpretation for this distance , but it does formally provide a reasonable definition of the geometric `` closeness '' for the two operators .  now suppose we have a definite prior state and choose . computing the weighted trace distance yields {\lvert i\rangle } ,\nonumber \\  & = { \langle i \rvert}{\hat{a}}^2{\lvert i\rangle } - \textstyle{\sum_f } { \left\lvert { \langle f \verti \rangle } \right\rvert}^2\,\left[\text{re}\frac{{\langle f \rvert}{\hat{a}}{\lvert i\rangle}}{{\langle f \vert i \rangle}}\right]^2 \nonumber \\  & \quad + \textstyle{\sum_f}{\left\lvert{ \langle f \vert i \rangle } \right\rvert}^2\,\left[\bar{a}_f - \text{re}\frac{{\langle f \rvert}{\hat{a}}{\lvert i\rangle}}{{\langle f \vert i \rangle}}\right]^2 .\end{aligned}\ ] ] only the final term depends on the choice of estimates for each , and is positive definite . therefore , the trace distance is minimized when this term vanishes , which in turn implies that the optimal estimate for each independent must be the weak value formula note that if the basis is chosen to be the eigenbasis of , then these optimal estimates reduce identically to the eigenvalues of and the trace distance vanishes .  to obtain eq .  including intermediate time - evolution , we can choose a particular and initial state that take into account hamiltonian propagation from the observed results for a specific preparation and postselection .this pure - state derivation may also be generalized in a straightforward way , which produces a formulation of the weak value suitable for generalized measurements and mixed states here the back - propagating operator is often called a `` retrodictive state , '' or `` effect matrix , '' in contrast to the forward - propagating `` predictive state '' , or `` density matrix . '' recently , the estimate in eq .  has been used to great effect experimentally for `` quantum smoothing '' and `` past quantum state '' analyses of continuously measured signals ( e.g. , it was used to track individual photon emissions into a monitored cavity ) .both and generally evolve according to open - system master equations that can also include the effects from additional ( discrete or continuous - in - time ) stochastic measurement - results , in contrast to the closed - system ( unitary ) schrdinger - von neumann dynamics usually assumed with eq . .note that if the effect matrix is the identity , then no posterior conditioning has been performed , so the usual expectation value is also recovered as a special case .   as a philosophical side note , for those who believe that the state - vector represents the complete physical ( ontic ) reality ( e.g. , adherents to the many - worlds interpretation ) , this time - symmetric estimate prompts several more radical speculations : the existence of the second state vector in eq.  seems to imply not only that the state is an incomplete description of reality at time , but also that there seems to be a causal effect on the time from the future time  .such a _ retro - causal _ interpretation is similar in spirit to the interpretations of anti - particles in quantum field theory as field - excitations that move backwards through time .however , just as with anti - particles , one does not need to invoke such controversial philosophical concepts as physical state - vectors or retro - causation to meaningfully interpret the weak value in eq .  as the best available statistical estimate given only the information about the specified boundary conditions .  a more pragmatic attitude ( which we shall adopt here )is to treat the estimate in eq .  as _ subjective _ ( epistemic ) , and pertaining to a time interval  ] ( see figure [ fig : disturb ] ) .supporting this point of view is the fact that similar bidirectional ( in time ) estimates about unknown properties of structured stochastic processes ( e.g. , hidden markov models ) during such an interval are now well - established in classical computational mechanics .there it is shown that one should use both forward and reverse `` causal states '' ( i.e. , probability distributions ) that contain information gathered both before _ and after _ each time to optimally estimate the properties of an evolving stochastic process . similarly ,classical statistics and filtering theory also use bidirectional states to provide the best estimate for information contained in noisy data confined to a time interval ( called optimally `` smoothing '' the noise ) .since quantum theory is closely related to probability theory , it is logical that similar estimation methods can be applied . indeed , upgrading these estimation schemes to the quantum realm produces both states in eq . , as well as the mixed - state generalization of eq .the confidence that estimations like the expectation value , or the weak value in eqs .  and , reflect something meaningful about the physical world ( and are not merely fevered hallucinations of the mind ) follows from verification of their predictions by experimental measurements . in the case of the expectation value ,any unbiased estimation of will suffice , corroborating the predicted result . in the case of the weak value, however , the presence of the posterior boundary condition additionally constrains the form of the possible measurements that can verify the estimate .specifically , those measurements must be `` weak , '' meaning that they should not appreciably perturb the evolution of the quantum system . since information extraction necessarily disturbs the quantum state , only minimally informative ( i.e., _ noisy _ ) measurements will leave the state mostly unperturbed , and thus faithfully reproduce the assumptions made about the evolution during the time interval  ] , and are the oscillation frequencies of the qubit and resonator , and is the dispersive frequency shift of the resonator that depends on the qubit state .note that the interaction term between the qubit and the resonator has the general von neumann form of eq . , but it is no longer impulsive .  assuming a pure state for the joint qubit - resonator system , we can make the following ansatz for the form of the joint state where are complex amplitudes , and are normalized resonator states that are correlated to each definite qubit state .it follows that the reduced qubit density matrix after tracing out the resonator has the following diagonal populations as well as an off - diagonal coherence that depends explicitly on the overlap between the two distinct and dynamically evolving resonator states that are correlated to definite qubit populations .the simple hamiltonian considered in eq .  is already diagonal in the energy basis of the qubit , so the populations in eq .do not change in time .however , the coherence in eq .  will display phase oscillations due to both the natural qubit energy splitting and the added influence of the dispersive resonator coupling .after extracting the components of the joint schrdinger equation by differentiating eq . , and a bit of algebra , it is straightforward to derive the evolution equation for the coherence directly from differentiating eq .\,\rho_{01}(t).\end{aligned}\ ] ] notably , the ( complex ) weak value of the resonator population naturally appears this weak value can be understood as a _ classical dynamical variable _ that completely determines the ensemble - averaged _ dynamical _ influence of the resonator on the reduced qubit state at each point in time , even in the absence of any explicit preselection or postselection measurements .the real part of this weak value is the best estimate of the population of the resonator mode if the qubit transitions between its ground and excited states at time . according to eq ., this weak value produces a shift of the natural qubit frequency , commonly known as the _ ac stark shift _notably , this shift does not involve either of the average mode populations or that one might naively expect , since the qubit coherence does not pertain to a definite population .any imaginary part of the weak value does not contribute to the ac stark shift in eq ., but instead produces a decay of the qubit coherence that indicates the resonator coupling is _ dephasing _ the qubit at a rate .  for an explicit example of this reduced evolution , to a good approximation a one - sided resonator with energy - decay rate ( as shown in figure [ fig : cqed ] ) that is pumped with a coherent state detuned by from the bare resonator frequency will reach a steady state that approximates the pure - state ansatz of eq . , with and approximating coherent states with complex classical amplitudes the weak value of the resonator population in eq.  correspondingly has the simple steady - state form ( assuming a wide - bandwidth resonator for brevity , with ) ,\end{aligned}\ ] ] which produces the ac stark shift of the qubit frequency with , as well as the dephasing rate .these expressions for the ac stark shift and ensemble - average dephasing rate agree with the known results for dispersive qubit measurements in circuit quantum electrodynamics ( cqed ) .we emphasize that the complex weak value of the resonator population in eq .is _ physically _ the relevant classical dynamical variable that controls the behavior of the ensemble - averaged reduced qubit evolution in eq . .indeed , the ac stark shift of the qubit frequency is the primary method used in cqed for extracting the average population in the resonator at steady state ( e.g. , ) , but we see here that in actuality such a dynamical method probes the _ weak value _ of that population , and not the population associated with any particular qubit state , or even the total average population .the weak value arises from the interference between the two fields and in the resonator that are correlated with the two definite qubit states , on average .  at each time the real part of indicates the best estimate of the average resonator excitation number seen by a qubit that is not in a definite energy eigenstate , while its imaginary part indicates the best estimate of the backaction on the qubit dynamics caused by the fluctuations of the resonator population around that average value .these latter fluctuations result in ensemble - average dephasing of the reduced qubit state , reinforcing the observation made for weak measurements that a weak value can only describes the average ( i.e. , classical mean field ) state of affairs for an ensemble of realizations . indeed ,when the time - dependent leakage from the resonator is accounted for , the qubit will not dephase in this manner , but will instead follow a pure _ quantum trajectory _ that depends on the leakage record . as such , we must necessarily interpret the weak value here as implicitly averaging over many such realizations in practice to produce a classical dynamical variable associated with the classical mean field in the resonator .given the consistent role of the real part of a weak value as a best conditioned observable estimate , we can now observe an intriguing logical tension inherent to the weak value .on one hand , any classical conditioned average must include disturbance to obtain anomalous values : the larger the disturbance , the more strange the average can become . on the other hand , the strangeness of the conditioned average in eq .is greatest when the quantum state is _ least _ disturbed by an intermediate measurement , and even persists when there is no added disturbance to the natural dynamical evolution in the example of eq . .these two statements imply that any classical ( hidden - variable ) explanation of a strange weak value as a disturbance effect must satisfy one of two properties : either ( a ) the quantum state must be a _ subjective _ ( epistemic ) quantity that is completely insensitive to whatever physical ( ontic ) disturbance is occurring , or ( b ) the relevant disturbance occurs entirely during the _ postselection _ , and not the intermediate measurement .classical fields that produce strange weak values in weak measurement experiments satisfy this second property , where the disturbance at the postselection filter causes interference between previously independent field components .however , quantum systems can display similar interference without wavelike intensities , and permit additional entanglement effects such as the dynamical evolution involving weak values emphasized in the previous section .especially for such a dynamical role of the weak value , it does not seem possible to ascribe strange weak values to classical disturbance mechanisms without also demanding that the quantum state is itself a fundamentally subjective collection of some more physical microstates ; such a demand , while not impossible , must contend with the pusey - barrett - rudolph theorem and the other no - go theorems ( reviewed in ) for states of such epistemic character .   to quantify this logical tension , we can express weak values in a more established and familiar way by rewriting eq .  using the spectral expansion to find , where this is a conditional _ quasiprobability _ distribution that weights the eigenvalues of in , and satisfies the normalization . as a result ,if a strange weak value is estimated , then at least one quasiprobability must be negative : .  since the conditioning denominator of eq .is positive - definite , we infer that the joint quasiprobability in the numerator of eq .[ also appearing directly in the experimental partial average of eq .] must be negative .this joint quasiprobability distribution is precisely the _ terletsky - margenau - hill distribution _ that has been used since the late 1930s .interestingly , the terletsky - margenau - hill distribution is the real part of the complex quasiprobability distribution introduced even earlier by kirkwood as an alternative to the wigner distribution .this kirkwood distribution is also known as the _ standard - ordering distribution _ for quantum phase space .in fact , dirac later considered this distribution specifically to discuss the classical - to - quantum transition , observing that the negativity arises directly from the usual operator noncommutativity of quantum mechanics .notably , the fully complex weak value that appears in reduced state dynamics is nothing more than a conditioned version of this complex kirkwood - dirac quasiprobability distribution .an important feature of the kirkwood - dirac distribution that has recently come to light is that any quantum state can be written in an operator basis such that this distribution forms its components .that is , if we define a suitable operator basis then we can write any quantum state using the expansion as such , the quasiprobabilities of the complex kirkwood - dirac distribution are a complete quantum state representation for an arbitrary density matrix that is analogous to a complex wavefunction for a pure state .unlike the usual wavefunction , however , the kirkwood - dirac distribution is directly compatible with bayes theorem [ as used in eq . ] , and thus behaves like a true probability distribution .this notable feature has enabled alternative methods of quantum state tomography by directly measuring complex weak values using von neumann interactions , and also permits fully bayesian quasiprobabilistic reformulations of coherent quantum dynamics .importantly , the negativity in such a quasiprobability representation of a quantum state is closely associated to traditional measures of nonclassicality .the usual examples of this criterion for nonclassicality are the negativity of the wigner distribution , or the glauber - sudarshan -distribution ; however , the terletsky - margenau - hill distribution in eq .has the same feature .  to emphasize this point, we can in fact relate weak values directly to the wigner distribution when we are considering infinite - dimensional systems . to see this , consider the wigner distribution for an initially pure state  where and represent the usual classical position and momentum variables .now suppose we compute the partial average of the momentum for a fixed using this joint quasiprobability distribution \int_{-\infty}^\infty\!\ !e^{ipy/\hbar}\frac{dp}{2\pi\hbar}dy , \nonumber \\  & \quad = \int_{-\infty}^\infty\!\ ! \frac{(-i\hbar\partial_x{\langle x-\frac{y}{2 } \vert i \rangle}){\langle i \vert x+\frac{y}{2 } \rangle } + \text{c.c.}}{2}\ , \delta(y)\,dy , \nonumber \\  & \quad = \text{re}(-i\hbar\partial_x{\langle x \vert i \rangle}){\langle i \vert x \rangle } , \nonumber \\  & \quad = \text{re}{\langle x\rvert}{\hat{p}}{\lvert i\rangle}{\langle i \vert x \rangle } , \nonumber \\  & \quad = \int_{-\infty}^\infty\!\ ! p\ , \text{re}{\langle x \vert p \rangle}{\langle p \vert i \rangle}{\langle i \vert x \rangle}\,dp . \nonumber\end{aligned}\ ] ] in other words , after performing integration - by - parts the partial average of the wigner function yields precisely the same result as averaging over the terletsky - margenau - hill distribution of eq . .it follows that conditioning this partial average directly produces a momentum weak value as the proper local average of momentum at the position  as with eq ., the denominator is a positive - definite probability , so only the partial average of eq .  in the numerator of eq .can produce nonclassical behavior .the inescapable conclusion is that weak values are intimately related to quasiprobability distributions for the quantum formalism in a fundamental and unavoidable way .moreover , these conditioned averages consistently confirm the best estimate that we expected from the derivation of eq . .similar relationships can be established for the other quantum phase space moyal distributions , of which the wigner function and the glauber - sudarshan -distribution are special - cases .( see for an example that emphasizes this point using non - symmetrized correlation functions . )we therefore have the following observation : if strange conditioned averages approximate the functional dependence of the weak value in eq ., then classical hidden variable models will be unable to satisfactorily explain that dependence .if one could , then it would also be able to reproduce other nonclassical statistical features of the quantum theory that arise from the negativity of these quasiprobability distributions ( see also ) .such negativity , in turn , arises from intrinsic quantum interference that is not present in classical systems of particles .   as a last , albeit poignant, illustration of the pervasive theoretical role of weak values in the quantum mechanical formalism , let us re - examine the schrdinger equation for a nonrelativistic particle {\lvert \psi\rangle}.\end{aligned}\ ] ] we can expand this equation into a wavefunction form that is local in the coordinate by contracting it with and rearranging this local form automatically involves the weak values of the kinetic and potential energies .  noting the familiar structure of this equation , we can then define a complex version of _ hamilton s principle function _ ( i.e. , the classical action ) as with real and imaginary parts that naturally isolate the phase ( i.e. , _ eikonal _ ) and probability density components of the wavefunction 12 & 12#1212_12%12[1][0] * * ,  ( ) * * ,  ( ) * * , ( ) * * ,  ( ) * * , ( ) * * ,  ( ) * * ,  ( ) `` , ''  ( ) ,  `` , ''  ( ) ,  `` , ''  ( ) ,  `` , ''  ( ) ,  `` , ''  ( ) ,  * * , ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) `` , ''  ( ) ,  * * , ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * , ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * , ( ) * * ,  ( ) * * ,  ( ) * * , ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * , ( ) * * , ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * , ( ) * * ,  ( ) * * ,  ( ) * * , ( ) * * ,  ( ) `` , ''  ( ) ,  `` , ''  ( ) ,  * * , ( ) * * ,  ( ) * * , ( ) * * ,  ( ) _ _ ( , ) _ _ ( , ) * * ,  ( ) * * , ( ) * * , ( ) `` ''  ( ) ,  * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) _ _ ( , ) link:\doibase 10.5772/2706 [ _ _ ] _ _ ( , ) * * ,  ( ) * * , ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) _ _ ( , ) * * ,  ( ) * * , ( ) * * ,  ( ) * * ,  ( ) _ _ ( , ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * , ( ) * * ,  ( ) in  _ _ , ( , ) chap ., p. ,  * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) link:\doibase  10.1038/ncomms4115 [ * * ,  ( ) ] * * ,  ( ) * * ,  ( ) _ _ ( , ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( )

Abstract: weak values arise experimentally as conditioned averages of weak ( noisy ) observable measurements that minimally disturb an initial quantum state , and also as dynamical variables for reduced quantum state evolution even in the absence of measurement . these averages can exceed the eigenvalue range of the observable ostensibly being estimated , which has prompted considerable debate regarding their interpretation . classical conditioned averages of noisy signals only show such anomalies if the quantity being measured is also disturbed prior to conditioning . this fact has recently been rediscovered , along with the question whether anomalous weak values are merely classical disturbance effects . here we carefully review the role of the weak value as both a conditioned observable estimation and a dynamical variable , and clarify why classical disturbance models will be insufficient to explain the weak value unless they can also simulate other quantum interference phenomena .


[ cols= " < , < , < " , ]the assumption that the relaxation modulus of a viscoelastic medium is a completely monotonic function has many implications for dispersion , attenuation and the kramers - kronig ( k - k ) dispersion relations .these aspects are very important for acoustics , in particular for the ultrasound applications .ultrasound attenuation and mechanical tests are used as complementary methods of investigating molecular relaxation in polymers and soft matter .interconversion of ultrasonic data and mechanical test data is important for improving the reliability of the results . in materials with dielectric propertiesmolecular relaxation can be investigated by studying dielectric loss .dielectric loss in polar dielectrics exhibits similar features ( alpha and beta peaks ) to the mechanical loss modulus while the dielectric permittivity , representing gradual polarization of the material in an electric field , has all the properties of mechanical creep .  in seismological applicationsthe minimum - phase aspect of viscoelastic wave propagation is relevant for deconvolution .the assumption that the relaxation modulus is a completely monotonic function has its roots in the interpolation of experimental data in terms of prony sums with positive coefficients . by bernsteins theorem complete monotonicity of the relaxation modulus is equivalent to the statement that the relaxation spectral measure is non - negative .a prony sum with positive coefficients is a completely monotonic function with a spectral measure supported by a finite set of points .several attempts have been made to justify complete monotonicity of the relaxation modulus by an assumption about dependence of stress on loops in the past history , by an assumed relation between time dependence of strain and stress or by other arguments .the theory of viscoelasticity based on the assumption of complete monotonicity of the relaxation modulus is very attractive because it leads to a fairly complete characterization of the creep compliance , complex modulus , attenuation and dispersion as well as the possible anisotropic properties of the medium . a weak point of the theory is an instability of the property of complete monotonicity : in an arbitrary neighborhood of a completely monotonic function in the space of bounded continuous functions there are functions which are not completely monotonic .the assumption of complete monotonicity has therefore to be considered as an a priori restriction on the data and on the interpolating functions .parameter estimation based on experimental data has to be performed in the class of completely monotonic functions .this is however implicit in routine modeling of experimental data for stress relaxation in terms of prony sums for general viscoelastic media or in terms of cole - cole , havriliak - negami and kohlrausch - williams - watts relaxation moduli for more specific classes of viscoelastic materials .the same relaxation functions are used in modeling dielectric relaxation in the same materials if they exhibit dielectric properties .completely monotonic relaxation moduli and the associated complex moduli are determined by a relaxation spectral measure , which represents the weight of participating debye relaxation mechanisms ( sec .[ sec : constitutive ] ) .it is shown below that the complex wave number function , the attenuation function and the dispersion function of a medium with a completely monotonic relaxation modulus are parameterized by the dispersion - attenuation spectral measure ( sec .[ sec : cwf ] ) .the dispersion function and the attenuation function satisfy a dispersion relation in parametric form because both functions are expressed in terms of a single spectral measure .this dispersion relation implies the k - k dispersion relations with two subtractions ( sec .[ sec : kk ] ) .the k - k dispersion relations indicate that the complex wave number function is the laplace transform of a causal distribution .this distribution turns out to be a second - order distributional derivative of a causal function defined in terms of the laplace transform of the dispersion - attenuation spectral measure .the k - k dispersion relations are closely related to the minimum phase property of viscoelastic green s functions ( sec .[ sec : mph ] ) .this property is relevant for deconvolution of seismic signals .the dispersion - attenuation spectral measure can often be explicitly calculated for a given analytic complex modulus by analytic continuation to the entire complex plane cut along the negative real semi - axis and calculating the jump of the analytic continuation at the branch cut . this procedure is demonstrated for the cole - cole , havriliak - negami and cole - davidson relaxation models ( sec .[ sec : examples ] ) .the dispersion - attenuation spectral measure provides an efficient tool for numerical determination of the dispersion and attenuation functions .a function defined on the open positive real semi - axis is said to be _ completely monotonic ( cm ) _ if it has derivatives of arbitrary high order and   a cm function can have a singularity at 0 . a bounded cm function has a limit at 0 and its domain of definition can therefore be extended by continuity to the closed positive real semi - axis .a cm function is locally integrable if and only if it is integrable over  ] and ,\infty [ } \mu({\mathrm{d}}r)/r < \infty ] and on , respectively . by a similar argumentis equivalent to the inequality ,\infty [ } \frac{\mu({\mathrm{d}}r)}{a + r } < \infty\ ] ] for an arbitrary number .a function defined on the closed positive real semi - axis is said to be a _ bernstein function_ if and is completely monotonic .  sincea bernstein function is non - decreasing and non - negative , the limit always exists and is finite .  a superposition of bernstein functions with positive coefficients is a bernstein function .the set of bernstein functions is closed under pointwise convergence .consequently an integral of a family of bernstein functions with a non - negative weight is a bernstein function .a function is a bernstein function if and only if it has the form where and is a licm function .indeed , is a cm function and , hence is a bernstein function . on the other hand ,if is a bernstein function then its derivative is a cm and it is locally integrable ..  holds with , .in particular the primitive function of the cm function is a bernstein function .[ def : cbf ] a function is a _ complete bernstein function _ ( cbf ) if there is a bernstein function such that .the set of complete bernstein functions will be denoted by the symbol .let be the bernstein function .+ the laplace transform of has the form , where and is a licm function . by bernstein s theorem ,where .hence if is a cbf and , then or , equivalently , ,\infty [ } \frac{\mu({\mathrm{d}}r)}{x + r } , \qquad x > 0\ ] ] where , and .any function with an integral representation of the form with and is obviously a cbf . in particular , the function is a cbf if .we now note that is a bernstein function because it is an integral of bernstein functions , , with a positive weight : eq .now implies that every cbf function is an integral of bernstein functions with a positive weight , hence it is a bernstein function .the bernstein function is however not a cbf . indeed , suppose the contrary. then has the integral representation with and . hence ,\infty [ }  \mu({\mathrm{d}}r)/(x + r) ] and therefore the characteristic function }(x ) = \int_{]0,\infty [ } { \mathrm{e}}^{-x y } \ , \mu({\mathrm{d}}y) ] is a square of the cbf function .a square of a cbf need not however be a cbf , for example is a cbf but its square is not even a bernstein function .[ thm : rebeta ] if then .if then ,\infty [ } \frac{\vert p \vert^2 + r\ , \re p}{\vert p + r\vert^2 } \nu({\mathrm{d}}r ) \geq 0\ ] ]  [ thm:2 ] is a non - decreasing function of in the closed right half plane .the integrand of eq .  has the form , where , , .the derivative of is non - negative .the derivative ,\infty [ } \frac{r\ , \nu({\mathrm{d}}r)}{(p + r)^2 } > 0\ ] ] exists for real because the integrand is bounded by .we are however interested in growth of as tends to infinity in the closed right complex half - plane .[ thm:2finitewavefrontspeed ] if then as in the right half of the complex -plane .the integrand of tends to 1 as . setting , the integrandis transformed into the function , which has a non - positive derivative .hence the integrand of is monotonically increasing to 1 .if , then by the lebesgue - fatou lemma  increases to infinity as , .[ thm : betaineq ] tends to 0 as in the right half - plane uniformly with respect to .  for  } \vert r\ , { \mathrm{e}}^{{\mathrm{i}}\varphi } + r\vert^2 = \inf_{\varphi \in [ -{\uppi}/2,{\uppi}/2 ] }  \left [ r^2 + r^2 + 2r r \cos(\varphi)\right ] = r^2 + r^2\ ] ] hence } \left\vert \frac{\beta\left(r { \mathrm{e}}^{{\mathrm{i}}\varphi}\right)}{r { \mathrm{e}}^{{\mathrm{i}}\varphi } } \right\vert \leq \int_{[0,\infty [ } \frac{\nu({\mathrm{d}}r)}{\sqrt{r^2 + r^2}}\ ] ] but for and the inequality holds in view of the inequality for , while for and , .thus for and .in view of eq . by the lebesgue dominated convergence theorem .hence tends to 0 in uniformly with respect to  ] and .it will be shown that green s function vanishes for if and thus can be identified as the _ wavefront speed_. the parameter , , will henceforth be replaced by varying in the range ..  implies that the function can be analytically continued to the complex plane cut along the negative real axis , i.e. to the principal riemann sheet , .the jump of the complex analytic function on the branch cut -\infty,0] ] is given by the formula , w ] ) = \frac{1}{{\uppi } } \int_u^w \im \lambda\left(r \ , { \mathrm{e}}^{{\mathrm{i}}{\uppi}}\right ) \ , { \mathrm{d}}r\ ] ] ( theorem [ thm : ancm ] ) .the function of a viscoelastic material satisfies the condition if the relaxation modulus is completely monotonic ..  implies that . in a viscoelastic medium with a completely monotonic relaxation modulusthe function is a cbf , hence by theorem [ thm : cbf2 ] is a cbf .  in the case of power law attenuation the function is a cbf if and only if .if then the exponent is negative and the function is not monotone , hence it is not a bf ; consequently the relaxation modulus is not completely monotonic .the function is not a possible candidate for the wave number function because is not a bernstein function .other counterexamples of this kind can be found in .we now examine the class of functions such that ..  implies that if and only if and .we shall use the following notation : for any real function defined on  where denotes the pointwise product of functions . if and only if .if the function is a cbf then the function is also a cbf and the complex wave number function ^{1/2} ] .the third factor is the square root of a cbf and hence a cbf itself . hence are also cbfs .we have thus proved the `` only if '' part .  for the converse we shall use the following identity ( theorem 7.11 in ) for ] with  the function is a non - decreasing function of in the right half complex -plane .it increases to infinity for in the right half complex -plane .if the spectral measure has infinite mass then the real part of the function increases to infinity in the right half complex -plane ( theorem [ thm:2finitewavefrontspeed ] ) .[ lem : finitewsp ] let .the function ^{-1}\ , { \mathrm{e}}^{-\beta(p)\ , r} ] .  by theorem [ thm : rebeta ] \vert ] .the integrand of the last integral is non - negative , hence } \frac{\mu({\mathrm{d}}r)}{(\vert p\vert + a)^2 } =  \frac{\vert p \vert^2}{(\vert p\vert + a)^2 } \mu([0,a])\ ] ] for an arbitrary positive ) ] and ) - \varepsilon_1 ] for all with and .  if , then the green s function vanishes for .  let .the green s function is given by an expression of the form , where is the bromwich contour running from to , , parallel to the imaginary axis , is a constant and the function is defined in lemma [ lem : finitewsp ] .consider the complex contour , with running from to and a fixed .lemma [ lem : finitewsp ] and jordan s lemma imply that the integral tends to 0 as .let be the straight line contour running from to . by the cauchy theorem the integral of over the contour vanishes .hence , taking the limit , the integral over vanishes .consequently for .  if then vanishes for .the thesis follows from eq . .a frequent feature of wave propagation in real viscoelastic media is wavefront smoothing . using a terminology often adopted in mechanics , many linear viscoelastic mediado not allow non - trivial discontinuity waves .it will now be shown that absence or presence of non - trivial discontinuity waves depends on a property of the dispersion - attenuation spectral measure .wavefront smoothing is ultimately due to the singularity of the derivative of the relaxation modulus at 0. we shall now relate it to the asymptotic properties of the attenuation - dispersion spectral measure .if all the moments of the spectral density function  are finite , then  .the dominating term is a positive constant .a special case is a finite bandwidth spectral density such as }(r) ] .if the function decays at an algebraic rate then some higher order moments are infinite and does not hold . assuming the asymptotic expansion of the spectral density the dissipation - attenuation function has the following asymptotic expansion at infinity  + r_n\ ] ] where  the asymptotic expansion of is valid in the entire cut complex plane ,  .the first term of the expansion of the dissipation - attenuation function is now while the attenuation function .consequently , if the spectral density decays algebraically at infinity , then vanishes asymptotically like .hence the integral is absolutely convergent and therefore the function is continuous .furthermore , the derivatives of are also given by absolutely convergent integrals .consequently is a smooth function of both arguments in . in particularit has continuous derivatives of arbitrary order at the wavefront .thus the wavefront does not carry any discontinuity of the green s function nor its derivatives .since the green s function vanishes for , it gradually decays to 0 with all its derivatives ..  implies that is also a smooth function of .the case of a strongly singular relaxation modulus is analyzed in theorem [ thm : strongsing ] .[ thm : strongsing ] if with and ) \sim_\infty r^\alpha \ ,l(r) ] , where , .if ) \sim_\infty r^\alpha \ , l(r) ] , with , , ,\infty[) ] where ,1[ ] is a non - increasing function of .therefore the phase speed is a non - decreasing function of frequency .eq .  andeq .  imply that   if then , in view of eq ., for all . the inverse of this limit will be denoted by the symbol . in particular , for , , this proves the following theorem :  [ thm : yyy ] if then and = 1/c_\infty + d ] is a function of regularly varying with index , ) \sim_\infty r^{1-\alpha}\ , l(r)\ ] ] where is a function of slow variation at infinity , then and   since , eq.  implies that .let ) ] , by valiron s theorem ( theorem [ thm : valiron ] ) this proves the theorem .the function in eq .is differentiable for , hence the attenuation and dispersion functions and are differentiable .the function by theorem [ thm : highfreq ] , hence it belongs to the space ( eq . ( 1.8.12 ) in sec .1.71.8 ) .consequently under the above hypotheses the functions and satisfy the k - k dispersion relations with one subtraction : where `` vp '' indicates that the integral is to be taken in the sense of principal value .low - frequency behavior of the attenuation function provides a useful test whether a specimen of the material subjected to constant strain for relaxes to zero stress - that is , whether .materials with vanishing are known as viscoelastic fluids .scalar models considered in this paper represent either longitudinal or shear waves .it turns out that the same material can behave under tension or compression like a viscoelastic solid and under shear strain as a viscoelastic fluid .if and the measure ) ] is a function of regularly varying at 0 with index , then .[ thm : lowfreq ] if the radon measure is regularly varying at 0 with ) = r^\gamma\ , l(r) ] .for example , if with , then with .the function is slowly varying at 0 , hence lemma [ thm : valiron ] implies the thesis .ad ( ii ) valiron s theorem implies the thesis .theorem [ thm : lowfreq ] is applicable to viscoelastic fluids . in the case of viscoelastic solidsimplies that and the hypotheses of theorem [ thm : lowfreq ] are not satisfied . in this case theorem [ thm : yyy ] implies that  ] , , then , where    if }(r ) \ , { \mathrm{d}}r ] at 0 : on the other hand , hence .the cole - cole model is defined by the laplace transform of the relaxation modulus of the form  .the cole - cole model and its generalizations are equivalent to linear viscoelastic models based on fractional derivatives considered in .the function is bounded completely monotonic ( appendix [ app : hncm ] ) . using the formula  the relaxation modulus can be expressed in terms of the mittag - leffler function \ ] ] the relaxation modulus has a finite limit at .the complex wave number function of the cole - cole model is given by the formula ^{1/2 } = b \ , p + p \int_{]0,\infty [ } \frac{\nu({\mathrm{d}}r)}{p + r}\ ] ] an explicit integral expression for the attenuation and dispersion functions of the cole - cole model can be derived from eqs ( [ eq : interepres][eq : interepres1 ] ) and . in the first place corollary [ cor : b ]implies that the wavefront speed equals .the spectral density of the cole - cole model is given by the formula = \frac{1}{{\uppi}\ , c_0 } \im \left [ \frac{1 + ( \tau r)^\alpha \ ,  \exp({\mathrm{i}}\alpha { \uppi})}{1 + a\ , ( \tau r)^\alpha \ , \exp({\mathrm{i}}\alpha { \uppi } ) }  \right]^{1/2}\ ] ] let where . the imaginary part of ^{1/2} ] .hence the spectral density is given by the expression where and .substitution of the expressions  where in the formulae yields the formula hence , using the identity and noting that the lower sign applies in the problem at hand , the final result is   note that and therefore .this implies that and { \mathrm{d}}r < \infty ] and thus . on the other hand ] .the lowest frequencies propagate with speeds close to and exhibit the attenuation  ] and hence consequently with .the low - frequency asymptotics of the cole - davidson attenuation function can be calculated by substituting eq .  in and using the definition of yields a behavior characteristic of a spectrum which does not extend to 0 : ] for some .the functions are analytic in and the function tends to zero as in .the inverse fourier transform is absolutely convergent and hence a continuous function of . by jordans lemma the function is causal .it is also a minimum phase distribution .let be a minimum phase tempered distribution .then satisfies the k - k dispersion relations with two subtractions .( sketchy ) by theorem [ thm : oup ] is an analytic function in .since is causal , by the paley - wiener theorem ( theorem [ thm : paley - wiener ] )  ] , the inverse fourier transform of is absolutely convergent , the function .hence is a distribution of second order and satisfies the k - k dispersion relations with two subtractions .the fourier transform of the green s function is the product of three factors the inverse fourier transform of is a convolution where is the inverse fourier transform of the first factor and is a minimum phase signal for each .the two first factors can be viewed as transfer functions or filters applied to the source signal .  since is a cbf function , the singularities of the function lie on the negative imaginary axis . by corollary [ cor : zeroscbf ]its zeros also lie on the negative imaginary axis .hence the first factor is a minimum phase filter .the third factor has two crucial properties  1 .it is the fourier transform of a causal tempered distribution ; 2 .the real and imaginary parts of the phase of the fourier transform of satisfy the k - k dispersion relations .these two properties identify as a minimum phase signal .note that by definition a minimum phase signal has an onset at zero time . in seismology and acoustics the onset time of the minimum phase signal can be arbitrary .the signal in is shifted and it starts at , the travel time to the point .the second factor in is an all - pass filter because it has a constant unit amplitude .it is a shift operator acting on the third factor .a shifted minimum phase function is again a minimum phase function .hence is a minimum phase signal .if the source function is a minimum phase signal , then the convolution is a minimum phase signal for every .we have demonstrated the utility of the spectral decomposition of the wave number function for investigating general properties of the attenuation and dispersion in a viscoelastic medium with a positive relaxation spectrum .it has also been shown that spectral decomposition of attenuation provides a very efficient method for numerical computation of attenuation in some popular viscoelastic models of polymers and soft matter .theorem [ thm : rebeta ] implies that the complex analytic continuation of the attenuation function , defined on the complex plane except on the negative imaginary axis , is a herglotz function .weaver and pao derived the k - k dispersion relations from this property .the derivation of the k - k dispersion relations adopted in this paper is based on a constitutive assumption .weaver and pao s assumption is not entirely based on a physical argument because it concerns the wave number function at complex - valued frequencies .41 natexlab#1#1[2]#2 , , , , . , , , ( ) . , , ( ), , , ( ) . , , , ( ) ., , , ( ) . , , , ( ) ., , , , ( ) . , , , ( ) . , , , , , , ., , , , . ., , , volume , , , ., , , , , ., , , , . , , , ( ) . , , , , , , ., , , ( ) . , , , ( ) . , , ( ) ., , , in : , , ( eds . ) , , , , , pp . . , , , , . , , , , , ( ) . , , , ( ) . , , , ( ) . , , , ( ) . , , , . , , ( ) . , , , , . , , , , , . , , ( ) . , , , , , ( ) . , , , ( ) . , , , , ( ) . , , , ( ) . , , , , , . , , ( ) . , , , , . , , volume of _ _ , , , . , , , , . , , , , , , .how can one verify whether a candidate function for the complex modulus is the fourier transform of a function vanishing for ?the best available answer to this question is theorem xii in :  [ thm : paley - wiener ] let the function be square integrable .+ if then there exists a function with support in the closed positive real semi - axis such that is the fourier transform of .the converse of this implication is also true .note that the hypotheses of the theorem involve conditions on the absolute value of only .a real function is said to be regularly varying at , where or , if is finite . is said to be rapidly varying with index if   a real function on is said to be rapidly varying at , where or , with index if   if is regularly varying at , where or , then , where is a real number .the number is called the index of the function at .  a real function on said to be slowly varying at if it is regularly varying at with index 0 .a regularly varying function with index at can be expressed as the product , where is slowly varying at .examples : is slowly varying at infinity ; is slowly varying at 0 if and regularly varying at infinity with index if ; is rapidly varying at infinity with index .( valiron 1911 , see ) [ thm : valiron ] if is an increasing function satisfying the condition and the function is given by the stieltjes integral then the following two statements are equivalent :  a.  ; b.    where and the function is slowly varying at infinity .valiron s theorem also holds when is replaced by 0 .[ thm : karamataabelianinfinity ] ( , chapter xiii ; , theorem 2.3 ) + if is non - decreasing right - continuous , is slowly varying at 0 , , and the laplace - stieltjes transform of converges for , where is a positive real number , then implies   note that the function is slowly varying at infinity .a non - negative real function on with the integral representation ,\infty [ } \frac{\mu({\mathrm{d}}r)}{x + r}\ ] ] where and , is called a _ stieltjes function_.  eq .can also be expressed in the form with .a stieltjes function has a complex analytic continuation in the complex plane cut along the negative real axis , .the function has the integral representation with replaced by .it follows immediately that maps to and vice versa .a stieltjes function has a finite value at every point of the complex plane outside the closure of the negative real axis .let .the inequality implies for the inequality ( cf remark [ rem:1 ] ) .if then we can find such a number that and .hence hence   [ thm : ancm ] + an analytic function regular in and satisfying the inequality ( i ) in and ( ii ) for can be expressed in the form where and .the limit exists and equals .the measure is uniquely defined as the weak limit for every continuous function with compact support .  [thm : invstcbf ] the inverse of a non - zero cbf is a stieltjes function .the integral representation implies that ,\infty [ } \frac{\rho({\mathrm{d}}y)}{1 + z\ , y } = a_1 + b \frac{1}{z } + \int_{]0,\infty [ }  \frac{u \,\rho\left({\mathrm{d}}\ , u^{-1}\right)}{u + y}\ ] ] where , .hence the function satisfies ( i ) and ( ii ) of theorem [ thm : ancm ] . in particular , ( ii ) implies that for .substitute .clearly , and , by reflection , for . since as well , is a stieltjes function .[ cor : zeroscbf ] a cbf does not vanish outside the closed negative real half - axis .let be a non - zero cbf .if then the stieltjes function has a pole at . hence .by the fubini theorem   if if integrable over [ 0,1 ] then the last integral is convergent .since the integrand is non - negative and the radon measure is positive , but hence the last integral is convergent .the integral is convergent , hence the inequality is satisfied .assume now that is satisfied .note that } \mu({\mathrm{d}}r ) \leq 2 \int_{[0,1 ] } \frac{\mu({\mathrm{d}}r)}{1 + r } < \infty\ ] ]it follows that the integral }  \frac{1 - { \mathrm{e}}^{-r}}{r } \ , \mu({\mathrm{d}}r)\ ] ] is convergent because the integrand is bounded .it remains to consider the integral on : hence is integrable over [ 0,1 ] .we prove here that is cm for . according to theorem 2.6 in chapter 16 of it suffices to prove that \geq 0 ] .the remaining hypotheses of theorem 2.6 are easy to verify .

Abstract: it is shown that viscoelastic wave dispersion and attenuation in a viscoelastic medium with a completely monotonic relaxation modulus is completely characterized by the phase speed and the dispersion - attenuation spectral measure . the dispersion and attenuation functions are expressed in terms of a single dispersion - attenuation spectral measure . an alternative expression of the mutual dependence of the dispersion and attenuation functions , known as the kramers - kronig dispersion relation , is also derived from the theory . the minimum phase aspect of the filters involved in the green s function is another consequence of the theory . explicit integral expressions for the attenuation and dispersion functions are obtained for a few analytical relaxation models .   viscoelasticity , wave propagation , completely monotonic , bernstein function , cole - cole model , havriliak - negami model , cole - davidson model 74d05 , 74j05 , 42a99