190 results found
Glashoff K, Bronstein MM, Asymptotic metrics on the space of matrices under the commutation relation
We show that the norm of the commutator defines "almost a metric" on thequotient space of commuting matrices, in the sense that it is a semi-metricsatisfying the triangle inequality asymptotically for large matrices drawn froma "good" distribution.
Glashoff K, Bronstein MM, Almost-commuting matrices are almost jointly diagonalizable
We study the relation between approximate joint diagonalization ofself-adjoint matrices and the norm of their commutator, and show that almostcommuting self-adjoint matrices are almost jointly diagonalizable by a unitarymatrix.
Bouritsas G, Bokhnyak S, Ploumpis S, et al., Neural 3D Morphable Models: Spiral Convolutional Networks for 3D Shape Representation Learning and Generation
Generative models for 3D geometric data arise in many important applicationsin 3D computer vision and graphics. In this paper, we focus on 3D deformableshapes that share a common topological structure, such as human faces andbodies. Morphable Models and their variants, despite their linear formulation,have been widely used for shape representation, while most of the recentlyproposed nonlinear approaches resort to intermediate representations, such as3D voxel grids or 2D views. In this work, we introduce a novel graphconvolutional operator, acting directly on the 3D mesh, that explicitly modelsthe inductive bias of the fixed underlying graph. This is achieved by enforcingconsistent local orderings of the vertices of the graph, through the spiraloperator, thus breaking the permutation invariance property that is adopted byall the prior work on Graph Neural Networks. Our operator comes by constructionwith desirable properties (anisotropic, topology-aware, lightweight,easy-to-optimise), and by using it as a building block for traditional deepgenerative architectures, we demonstrate state-of-the-art results on a varietyof 3D shape datasets compared to the linear Morphable Model and other graphconvolutional operators.
Eynard D, Glashoff K, Bronstein MM, et al., Multimodal diffusion geometry by joint diagonalization of Laplacians
We construct an extension of diffusion geometry to multiple modalitiesthrough joint approximate diagonalization of Laplacian matrices. This naturallyextends classical data analysis tools based on spectral geometry, such asdiffusion maps and spectral clustering. We provide several synthetic and realexamples of manifold learning, retrieval, and clustering demonstrating that thejoint diffusion geometry frequently better captures the inherent structure ofmulti-modal data. We also show that many previous attempts to constructmultimodal spectral clustering can be seen as particular cases of jointapproximate diagonalization of the Laplacians.
Bronstein MM, Multimodal diff-hash
Many applications require comparing multimodal data with different structureand dimensionality that cannot be compared directly. Recently, there has beenincreasing interest in methods for learning and efficiently representing suchmultimodal similarity. In this paper, we present a simple algorithm formultimodal similarity-preserving hashing, trying to map multimodal data intothe Hamming space while preserving the intra- and inter-modal similarities. Weshow that our method significantly outperforms the state-of-the-art method inthe field.
Bronstein MM, Kernel diff-hash
This paper presents a kernel formulation of the recently introduced diff-hashalgorithm for the construction of similarity-sensitive hash functions. Ourkernel diff-hash algorithm that shows superior performance on the problem ofimage feature descriptor matching.
Bronstein AM, Spectral descriptors for deformable shapes
Informative and discriminative feature descriptors play a fundamental role indeformable shape analysis. For example, they have been successfully employed incorrespondence, registration, and retrieval tasks. In the recent years,significant attention has been devoted to descriptors obtained from thespectral decomposition of the Laplace-Beltrami operator associated with theshape. Notable examples in this family are the heat kernel signature (HKS) andthe wave kernel signature (WKS). Laplacian-based descriptors achievestate-of-the-art performance in numerous shape analysis tasks; they arecomputationally efficient, isometry-invariant by construction, and cangracefully cope with a variety of transformations. In this paper, we formulatea generic family of parametric spectral descriptors. We argue that in order tobe optimal for a specific task, the descriptor should take into account thestatistics of the corpus of shapes to which it is applied (the "signal") andthose of the class of transformations to which it is made insensitive (the"noise"). While such statistics are hard to model axiomatically, they can belearned from examples. Following the spirit of the Wiener filter in signalprocessing, we show a learning scheme for the construction of optimal spectraldescriptors and relate it to Mahalanobis metric learning. The superiority ofthe proposed approach is demonstrated on the SHREC'10 benchmark.
Kovnatsky A, Bronstein MM, Bronstein AM, et al., Diffusion framework for geometric and photometric data fusion in non-rigid shape analysis
In this paper, we explore the use of the diffusion geometry framework for thefusion of geometric and photometric information in local and global shapedescriptors. Our construction is based on the definition of a diffusion processon the shape manifold embedded into a high-dimensional space where theembedding coordinates represent the photometric information. Experimentalresults show that such data fusion is useful in coping with differentchallenges of shape analysis where pure geometric and pure photometric methodsfail.
Bronstein AM, Bronstein MM, Kimmel R, The Video Genome
Fast evolution of Internet technologies has led to an explosive growth ofvideo data available in the public domain and created unprecedented challengesin the analysis, organization, management, and control of such content. Theproblems encountered in video analysis such as identifying a video in a largedatabase (e.g. detecting pirated content in YouTube), putting together videofragments, finding similarities and common ancestry between different versionsof a video, have analogous counterpart problems in genetic research andanalysis of DNA and protein sequences. In this paper, we exploit the analogybetween genetic sequences and videos and propose an approach to video analysismotivated by genomic research. Representing video information as video DNAsequences and applying bioinformatic algorithms allows to search, match, andcompare videos in large-scale databases. We show an application forcontent-based metadata mapping between versions of annotated video.
Sprechmann P, Bronstein AM, Sapiro G, Learning Robust Low-Rank Representations
In this paper we present a comprehensive framework for learning robustlow-rank representations by combining and extending recent ideas for learningfast sparse coding regressors with structured non-convex optimizationtechniques. This approach connects robust principal component analysis (RPCA)with dictionary learning techniques and allows its approximation via trainableencoders. We propose an efficient feed-forward architecture derived from anoptimization algorithm designed to exactly solve robust low dimensionalprojections. This architecture, in combination with different trainingobjective functions, allows the regressors to be used as online approximants ofthe exact offline RPCA problem or as RPCA-based neural networks. Simplemodifications of these encoders can handle challenging extensions, such as theinclusion of geometric data transformations. We present several examples withreal data from image, audio, and video processing. When used to approximateRPCA, our basic implementation shows several orders of magnitude speedupcompared to the exact solvers with almost no performance degradation. We showthe strength of the inclusion of learning to the RPCA approach on a musicsource separation application, where the encoders outperform the exact RPCAalgorithms, which are already reported to produce state-of-the-art results on abenchmark database. Our preliminary implementation on an iPad showsfaster-than-real-time performance with minimal latency.
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