A dual framework for low-rank tensor completion

One of the popular approaches for low-rank tensor completion is to use the latent trace norm regularization. However, most existing works in this direction learn a sparse combination of tensors. In this work, we fill this gap by proposing a variant of the latent trace norm that helps in learning a non-sparse combination of tensors. We develop a dual framework for solving the low-rank tensor completion problem. We first show a novel characterization of the dual solution space with an interesting factorization of the optimal solution. Overall, the optimal solution is shown to lie on a Cartesian product of Riemannian manifolds. Furthermore, we exploit the versatile Riemannian optimization framework for proposing computationally efficient trust region algorithm. The experiments illustrate the efficacy of the proposed algorithm on several real-world datasets across applications.Comment: Aceepted to appear in Advances of Nueral Information Processing Systems (NIPS), 2018. A shorter version appeared in the NIPS workshop on Synergies in Geometric Data Analysis 201

Impossibility of dimension reduction in the nuclear norm

Let $\mathsf{S}_1$ (the Schatten--von Neumann trace class) denote the Banach space of all compact linear operators $T:\ell_2\to \ell_2$ whose nuclear norm $\|T\|_{\mathsf{S}_1}=\sum_{j=1}^\infty\sigma_j(T)$ is finite, where $\{\sigma_j(T)\}_{j=1}^\infty$ are the singular values of $T$. We prove that for arbitrarily large $n\in \mathbb{N}$ there exists a subset $\mathcal{C}\subseteq \mathsf{S}_1$ with $|\mathcal{C}|=n$ that cannot be embedded with bi-Lipschitz distortion $O(1)$ into any $n^{o(1)}$-dimensional linear subspace of $\mathsf{S}_1$. $\mathcal{C}$ is not even a $O(1)$-Lipschitz quotient of any subset of any $n^{o(1)}$-dimensional linear subspace of $\mathsf{S}_1$. Thus, $\mathsf{S}_1$ does not admit a dimension reduction result \'a la Johnson and Lindenstrauss (1984), which complements the work of Harrow, Montanaro and Short (2011) on the limitations of quantum dimension reduction under the assumption that the embedding into low dimensions is a quantum channel. Such a statement was previously known with $\mathsf{S}_1$ replaced by the Banach space $\ell_1$ of absolutely summable sequences via the work of Brinkman and Charikar (2003). In fact, the above set $\mathcal{C}$ can be taken to be the same set as the one that Brinkman and Charikar considered, viewed as a collection of diagonal matrices in $\mathsf{S}_1$. The challenge is to demonstrate that $\mathcal{C}$ cannot be faithfully realized in an arbitrary low-dimensional subspace of $\mathsf{S}_1$, while Brinkman and Charikar obtained such an assertion only for subspaces of $\mathsf{S}_1$ that consist of diagonal operators (i.e., subspaces of $\ell_1$). We establish this by proving that the Markov 2-convexity constant of any finite dimensional linear subspace $X$ of $\mathsf{S}_1$ is at most a universal constant multiple of $\sqrt{\log \mathrm{dim}(X)}$

Non-Abelian Analogs of Lattice Rounding

Lattice rounding in Euclidean space can be viewed as finding the nearest point in the orbit of an action by a discrete group, relative to the norm inherited from the ambient space. Using this point of view, we initiate the study of non-abelian analogs of lattice rounding involving matrix groups. In one direction, we give an algorithm for solving a normed word problem when the inputs are random products over a basis set, and give theoretical justification for its success. In another direction, we prove a general inapproximability result which essentially rules out strong approximation algorithms (i.e., whose approximation factors depend only on dimension) analogous to LLL in the general case.Comment: 30 page

Deep Grassmann Manifold Optimization for Computer Vision

In this work, we propose methods that advance four areas in the field of computer vision: dimensionality reduction, deep feature embeddings, visual domain adaptation, and deep neural network compression. We combine concepts from the fields of manifold geometry and deep learning to develop cutting edge methods in each of these areas. Each of the methods proposed in this work achieves state-of-the-art results in our experiments. We propose the Proxy Matrix Optimization (PMO) method for optimization over orthogonal matrix manifolds, such as the Grassmann manifold. This optimization technique is designed to be highly flexible enabling it to be leveraged in many situations where traditional manifold optimization methods cannot be used. We first use PMO in the field of dimensionality reduction, where we propose an iterative optimization approach to Principal Component Analysis (PCA) in a framework called Proxy Matrix optimization based PCA (PM-PCA). We also demonstrate how PM-PCA can be used to solve the general $L_p$-PCA problem, a variant of PCA that uses arbitrary fractional norms, which can be more robust to outliers. We then present Cascaded Projection (CaP), a method which uses tensor compression based on PMO, to reduce the number of filters in deep neural networks. This, in turn, reduces the number of computational operations required to process each image with the network. Cascaded Projection is the first end-to-end trainable method for network compression that uses standard backpropagation to learn the optimal tensor compression. In the area of deep feature embeddings, we introduce Deep Euclidean Feature Representations through Adaptation on the Grassmann manifold (DEFRAG), that leverages PMO. The DEFRAG method improves the feature embeddings learned by deep neural networks through the use of auxiliary loss functions and Grassmann manifold optimization. Lastly, in the area of visual domain adaptation, we propose the Manifold-Aligned Label Transfer for Domain Adaptation (MALT-DA) to transfer knowledge from samples in a known domain to an unknown domain based on cross-domain cluster correspondences

Fisher and Kernel Fisher Discriminant Analysis: Tutorial

This is a detailed tutorial paper which explains the Fisher discriminant Analysis (FDA) and kernel FDA. We start with projection and reconstruction. Then, one- and multi-dimensional FDA subspaces are covered. Scatters in two- and then multi-classes are explained in FDA. Then, we discuss on the rank of the scatters and the dimensionality of the subspace. A real-life example is also provided for interpreting FDA. Then, possible singularity of the scatter is discussed to introduce robust FDA. PCA and FDA directions are also compared. We also prove that FDA and linear discriminant analysis are equivalent. Fisher forest is also introduced as an ensemble of fisher subspaces useful for handling data with different features and dimensionality. Afterwards, kernel FDA is explained for both one- and multi-dimensional subspaces with both two- and multi-classes. Finally, some simulations are performed on AT&T face dataset to illustrate FDA and compare it with PCA

Noncommutative geometry and motives: the thermodynamics of endomotives

We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of $L$-functions. The analogue in characteristic zero of the action of the Frobenius on l-adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitable category of motives) of a restriction map of noncommutative spaces. The latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive (a noncommutative generalization of Artin motives). Semigroups of endomorphisms of algebraic varieties give rise canonically to such endomotives, with an action of the absolute Galois group. The semigroup of endomorphisms of the multiplicative group yields the Bost-Connes system, from which one obtains, through the above procedure, the desired cohomological interpretation of the zeros of the Riemann zeta function. In the last section we also give a Lefschetz formula for the archimedean local L-factors of arithmetic varieties.Comment: 52 pages, amslatex, 1 eps figure, v2: final version to appea