We study the problem of supervised learning a metric space under discriminative constraints. Given a universe @math and sets @math of similar and dissimilar pairs, we seek to find a mapping @math , into some target metric space @math , such that similar objects are mapped to points at distance at most @math , and dissimilar objects are mapped to points at distance at least @math . More generally, the goal is to find a mapping of maximum accuracy (that is, fraction of correctly classified pairs). We propose approximation algorithms for various versions of this problem, for the cases of Euclidean and tree metric spaces. For both of these target spaces, we obtain fully polynomial-time approximation schemes (FPTAS) for the case of perfect information. In the presence of imperfect information we present approximation algorithms that run in quasipolynomial time (QPTAS). Our algorithms use a combination of tools from metric embeddings and graph partitioning, that could be of independent interest.
We note that other notions of supervision have also been considered in the metric learning literature. For example, instead of pairs of objects that are labeled either similar'' or dissimilar'', another possibility is to have triple constraints of the form @math , encoding the fact that @math is more similar to @math than @math ''. In this case, one seeks to find a mapping @math , such that @math , for some parameter @math (see e.g. @cite_22 ). However, the form of supervision that we consider is one of the most popular ones in practice. Furthermore, it is often desirable that the mapping @math has a special form, such as linearity (when @math is a subset of some linear space), or being specified implicitly via a set of parameters, as is the case of mappings computed by neural networks. We believe that our work can lead to further theoretical understanding of the metric learning problem under different types of supervision, and for specific classes of embeddings.
@cite_22
2605102252 2142863987 2068042582 219040644
We address the problem of distance metric learning (DML), defined as learning a distance consistent with a notion of semantic similarity. Traditionally, for this problem supervision is expressed in the form of sets of points that follow an ordinal relationship – an anchor point x is similar to a set of positive points Y , and dissimilar to a set of negative points Z, and a loss defined over these distances is minimized. While the specifics of the optimization differ, in this work we collectively call this type of supervision Triplets and all methods that follow this pattern Triplet-Based methods. These methods are challenging to optimize. A main issue is the need for finding informative triplets, which is usually achieved by a variety of tricks such as increasing the batch size, hard or semi-hard triplet mining, etc. Even with these tricks, the convergence rate of such methods is slow. In this paper we propose to optimize the triplet loss on a different space of triplets, consisting of an anchor data point and similar and dissimilar proxy points which are learned as well. These proxies approximate the original data points, so that a triplet loss over the proxies is a tight upper bound of the original loss. This proxy-based loss is empirically better behaved. As a result, the proxy-loss improves on state-of-art results for three standard zero-shot learning datasets, by up to 15 points, while converging three times as fast as other triplet-based losses. Metric learning aims at finding a distance that approximates a task-specific notion of semantic similarity. Typically, a Mahalanobis distance is learned from pairs of data labeled as being semantically similar or not. In this paper, we learn such metrics in a weakly supervised setting where "bags" of instances are labeled with "bags" of labels. We formulate the problem as a multiple instance learning (MIL) problem over pairs of bags. If two bags share at least one label, we label the pair positive, and negative otherwise. We propose to learn a metric using those labeled pairs of bags, leading to MildML, for multiple instance logistic discriminant metric learning. MildML iterates between updates of the metric and selection of putative positive pairs of examples from positive pairs of bags. To evaluate our approach, we introduce a large and challenging data set, Labeled Yahoo! News, which we have manually annotated and contains 31147 detected faces of 5873 different people in 20071 images. We group the faces detected in an image into a bag, and group the names detected in the caption into a corresponding set of labels. When the labels come from manual annotation, we find that MildML using the bag-level annotation performs as well as fully supervised metric learning using instance-level annotation. We also consider performance in the case of automatically extracted labels for the bags, where some of the bag labels do not correspond to any example in the bag. In this case MildML works substantially better than relying on noisy instance-level annotations derived from the bag-level annotation by resolving face-name associations in images with their captions. In this paper, we raise important issues on scalability and the required degree of supervision of existing Mahalanobis metric learning methods. Often rather tedious optimization procedures are applied that become computationally intractable on a large scale. Further, if one considers the constantly growing amount of data it is often infeasible to specify fully supervised labels for all data points. Instead, it is easier to specify labels in form of equivalence constraints. We introduce a simple though effective strategy to learn a distance metric from equivalence constraints, based on a statistical inference perspective. In contrast to existing methods we do not rely on complex optimization problems requiring computationally expensive iterations. Hence, our method is orders of magnitudes faster than comparable methods. Results on a variety of challenging benchmarks with rather diverse nature demonstrate the power of our method. These include faces in unconstrained environments, matching before unseen object instances and person re-identification across spatially disjoint cameras. In the latter two benchmarks we clearly outperform the state-of-the-art. Is strong supervision necessary for learning a good visual representation? Do we really need millions of semantically-labeled images to train a Convolutional Neural Network (CNN)? In this paper, we present a simple yet surprisingly powerful approach for unsupervised learning of CNN. Specifically, we use hundreds of thousands of unlabeled videos from the web to learn visual representations. Our key idea is that visual tracking provides the supervision. That is, two patches connected by a track should have similar visual representation in deep feature space since they probably belong to same object or object part. We design a Siamese-triplet network with a ranking loss function to train this CNN representation. Without using a single image from ImageNet, just using 100K unlabeled videos and the VOC 2012 dataset, we train an ensemble of unsupervised networks that achieves 52 mAP (no bounding box regression). This performance comes tantalizingly close to its ImageNet-supervised counterpart, an ensemble which achieves a mAP of 54.4 . We also show that our unsupervised network can perform competitively in other tasks such as surface-normal estimation.