arXiv research

In recent years, a class of dictionaries have been proposed for multidimensional (tensor) data representation that exploit the structure of tensor data by imposing a Kronecker structure on the dictionary underlying the data. In this work, a novel algorithm called "STARK" is provided to learn Kronecker structured dictio…

Four algorithms improve sparse tensor BR1Approx with theoretical guarantees.

A new algorithm completes rank-1 tensors with minimal samples and time.

Study explores warped geometries of tensor manifolds, finding non-geodesic connections for some parameters.

New compact ECS manifolds with rank 2 discovered, differing from previous rank 1 examples.

In this paper, we provide local and global convergence guarantees for recovering CP (Candecomp/Parafac) tensor decomposition. The main step of the proposed algorithm is a simple alternating rank-$1$ update which is the alternating version of the tensor power iteration adapted for asymmetric tensors. Local convergence g…

GETF efficiently decomposes large-scale Boolean tensors.

In this paper we propose a tensor-based nonlinear model for high-order data classification. The advantages of the proposed scheme are that (i) it significantly reduces the number of weight parameters, and hence of required training samples, and (ii) it retains the spatial structure of the input samples. The proposed mo…

We study stability and local minimizing properties of $L^p$- norms of Riemannian curvature tensor denoted by $\mathcal{R}_p$ by variational methods. We compute the Hessian of $\mathcal{R}_p$ at compact rank 1 symmetric spaces and prove that they are stable for $\mathcal{R}_p$ for certain values of p > 2. A similar resu…

We study rank-1 {L1-norm-based TUCKER2} (L1-TUCKER2) decomposition of 3-way tensors, treated as a collection of $N$ $D \times M$ matrices that are to be jointly decomposed. Our contributions are as follows. i) We prove that the problem is equivalent to combinatorial optimization over $N$ antipodal-binary variables. ii)…

Tensor PCA problem analyzed with statistical query lower bounds.

Tensor decompositions have rich applications in statistics and machine learning, and developing efficient, accurate algorithms for the problem has received much attention recently. Here, we present a new method built on Kruskal's uniqueness theorem to decompose symmetric, nearly orthogonally decomposable tensors. Unlik…

New findings on tensor decomposition complexity, showing polynomial functions can estimate the largest component under certain conditions.

The paper analyzes deflation for estimating a low-rank spike in large tensors with noise.

Efficiently reduces tensor ranks using mean-field approximation.

Study on rank-one ECS manifolds, focusing on dilational type.

TSL learns separable models to avoid signal cancellation and off-support extrapolation.

We give two characterizations of varieties whose universal cover is a bounded symmetric domain without ball factors in terms of the existence of a holomorphic endomorphism \s of the tensor product T\otimes T' of the tangent bundle T with the cotangent bundle T'. To such a curvature type tensor \s one associates the fir…

Study analyzes accuracy of tensor deflation in noisy conditions.

By considering the projectivized spectrum of the Jacobi operator, we introduce the concept of projective Osserman manifold in both the affine and in the pseudo-Riemannian settings. If M is an affine projective Osserman manifold, then the modified Riemannian extension metric on the cotangent bundle is both spacelike and…

We analyze low rank tensor completion (TC) using noisy measurements of a subset of the tensor. Assuming a rank-$r$, order-$d$, $N \times N \times \cdots \times N$ tensor where $r=O(1)$, the best sampling complexity that was achieved is $O(N^{\frac{d}{2}})$, which is obtained by solving a tensor nuclear-norm minimizatio…

Paper studies tensor models using random matrix theory.

PSMM method optimizes matrix sufficient dimension reduction.

Fourier PCA is Principal Component Analysis of a matrix obtained from higher order derivatives of the logarithm of the Fourier transform of a distribution.We make this method algorithmic by developing a tensor decomposition method for a pair of tensors sharing the same vectors in rank-$1$ decompositions. Our main appli…

In this paper we study the problem of learning the weights of a deep convolutional neural network. We consider a network where convolutions are carried out over non-overlapping patches with a single kernel in each layer. We develop an algorithm for simultaneously learning all the kernels from the training data. Our app…

Proves conditions for Fourier transforms in rank 1 symmetric spaces.

Study shows dynamics of rank 1 orbifolds in flat surfaces.

Locally conformally Hessian manifolds are dense in radiant ones of rank 1.

Estimates rank-one spikes from heavy-tailed noise using self-avoiding walks.

Sharp isoperimetric inequalities for Neumann eigenvalues in symmetric spaces.

DTCCA learns nonlinear transformations of multi-view data for high-order correlation.

Right inverse found for Cartan differential in rank-1 symmetric spaces.

A1GM method improves efficiency in reconstructing missing data using KL divergence.

The paper describes hyperkähler geometry of cotangent bundles using rank-1 projections.

Consider an action of a connected compact Lie group on a compact complex manifold $M$, and two equivariant vector bundles $L$ and $E$ on $M$, with $L$ of rank 1. The purpose of this paper is to establish holomorphic Morse inequalities à la Demailly for the invariant part of the Dolbeault cohomology of tensor powers of …

Volume comparison theorem for rank 1 symmetric spaces proved.

Paper presents a rank-1 approximation method for natural policy gradients in deep RL.

Improved sample and time complexity for identifying mixtures of product distributions.

Rank-1 BNNs improve efficiency and scalability of Bayesian neural nets.

Study on diagonal and separating coordinates for symmetric spaces of rank 1.

If $\mathcal{M}=(M,\nabla)$ is an affine surface, let $\mathcal{Q}(\mathcal{M}):=\ker(\mathcal{H}+\frac1{m-1}ρ_s)$ be the space of solutions to the quasi-Einstein equation for the crucial eigenvalue. Let $\tilde{\mathcal{M}}=(M,\tilde\nabla)$ be another affine structure on $M$ which is strongly projectively flat. We sh…

We establish the proportionality principle between the Riemannian volume and locally finite simplicial volume for Q-rank 1 locally symmetric spaces covered by products of hyperbolic spaces, giving the first examples for manifolds whose cusp groups are not necessarily amenable. Also, we give a simple direct proof of the…

We use generalised cross--ratios to prove the Ptolemaean inequality and the Theorem of Ptolemaeus in the setting of the boundary of symmetric Riemannian spaces of rank 1 and of negative curvature.

For a given lattice, we establish an equivalence involving a closed zone of the corresponding Voronoi polytope, a lamina hyperplane of the corresponding Delaunay partition and a quadratic form of rank 1 being an extreme ray of the corresponding L-type domain.

We study rank $1$ flat bundles over solvmanifolds whose cohomologies are non-trivial. By using Hodge theoretical properties for all topologically trivial rank $1$ flat bundles, we represent the structure theorem of Kähler solvmanifolds as extensions of Hasegawa's result and Benson-Gordon's result for nilmanifolds.

We consider the heat kernel (and the zeta function) associated with Laplace type operators acting on a general irreducible rank 1 locally symmetric space X. The set of Minakshisundaram- Pleijel coefficients {A_k(X)}_{k=0}^{\infty} in the short-time asymptotic expansion of the heat kernel is calculated explicitly.

We study the critical points of the renormalized volume for acylindrical geometrically finite hyperbolic 3-manifolds that include rank-1 cusps, and show that the renormalized volume is locally convex around these critical points. We give a modified definition of the renormalized volume that is additive under gluing, an…

We explore the algebraic structure of the solution space of convex optimization problem Constrained Minimum Trace Factor Analysis (CMTFA), when the population covariance matrix $Σ_x$ has an additional latent graphical constraint, namely, a latent star topology. In particular, we have shown that CMTFA can have either a …