Mapper is an unsupervised machine learning algorithm generalising the notion of clustering to obtain a geometric description of a dataset. The procedure splits the data into possibly overlapping bins which are then clustered. The output of the algorithm is a graph where nodes represent clusters and edges represent the sharing of data points between two clusters. However, several parameters must be selected before applying Mapper and the resulting graph may vary dramatically with the choice of parameters. We define an intrinsic notion of Mapper instability that measures the variability of the output as a function of the choice of parameters required to construct a Mapper output. Our results and discussion are general and apply to all Mapper-type algorithms. We derive theoretical results that provide estimates for the instability and suggest practical ways to control it. We provide also experiments to illustrate our results and in particular we demonstrate that a reliable candidate Mapper output can be identified as a local minimum of instability regarded as a function of Mapper input parameters.
Dey, M ' e moli and Wang @cite_13 @cite_30 study the structure and stability of a stable signature for what they called , which uses a hierarchy of covers instead of a single one. However, it is not clear how to translate their findings to the context of the original Mapper.
@cite_30 @cite_13
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Dept. of Computer Science, K.U.LeuvenCelestijnenlaan 200A, B-3001 Heverlee, BelgiumE-mail: pelov,marcd,maurice @cs.kuleuven.ac.beAbstract. We deﬁne a translation of aggregate programs to normal lo-gic programs which preserves the set of partial stable models. We thendeﬁne the classes of deﬁnite and stratiﬁed aggregate programs and showthat the translation of such programs are, respectively, deﬁnite and strat-iﬁed logic programs. Consequently these two classes of programs havea single partial stable model which is two-valued and is also the well-founded model. Our deﬁnition of stratiﬁcation is more general than theexisting one and covers a strictly larger class of programs. This article improves recent methods for large scale image search. We first analyze the bag-of-features approach in the framework of approximate nearest neighbor search. This leads us to derive a more precise representation based on Hamming embedding (HE) and weak geometric consistency constraints (WGC). HE provides binary signatures that refine the matching based on visual words. WGC filters matching descriptors that are not consistent in terms of angle and scale. HE and WGC are integrated within an inverted file and are efficiently exploited for all images in the dataset. We then introduce a graph-structured quantizer which significantly speeds up the assignment of the descriptors to visual words. A comparison with the state of the art shows the interest of our approach when high accuracy is needed. Experiments performed on three reference datasets and a dataset of one million of images show a significant improvement due to the binary signature and the weak geometric consistency constraints, as well as their efficiency. Estimation of the full geometric transformation, i.e., a re-ranking step on a short-list of images, is shown to be complementary to our weak geometric consistency constraints. Our approach is shown to outperform the state-of-the-art on the three datasets. We present an offline signature verification system using three different pseudo-dynamic features, two different classifier training approaches and two datasets. One of the most difficult problems of off-line signature verification is that the signature is just a static image while losing a lot of useful dynamic information. Three separate pseudo-dynamic features based on gray level: local binary pattern (LBP), gray level co-occurrence matrix (GLCM) and histogram of oriented gradients (HOG) are used. The classification is performed using writer-dependent Support Vector Machine (SVMs) classifiers and Global Real Adaboost method, where two different approaches to train the classifier. In the first mode, each SVM is trained with the feature vectors obtained from the reference signatures of the corresponding user and those random forgeries for each signer while the global Adaboost classifier is trained using genuine and random forgery signatures of signers that are excluded from the test set. The fusion of all features achieves the best result of 7.66 and 9.94 equal error rate in GPDS while 7.55 and 11.55 equal error rate in CSD respectively. Given a continuous function f:X->R and a cover I of its image by intervals, the Mapper is the nerve of a refinement of the pullback cover f^ -1 (I). Despite its success in applications, little is known about the structure and stability of this construction from a theoretical point of view. As a pixelized version of the Reeb graph of f, it is expected to capture a subset of its features (branches, holes), depending on how the interval cover is positioned with respect to the critical values of the function. Its stability should also depend on this positioning. We propose a theoretical framework relating the structure of the Mapper to that of the Reeb graph, making it possible to predict which features will be present and which will be absent in the Mapper given the function and the cover, and for each feature, to quantify its degree of (in-)stability. Using this framework, we can derive guarantees on the structure of the Mapper, on its stability, and on its convergence to the Reeb graph as the granularity of the cover I goes to zero.