New method solves rank-1 L1-norm TUCKER2 decomposition efficiently.
problem Exact solution to rank-1 L1-norm TUCKER2 decomposition of tensors.
method Proved equivalent to combinatorial optimization, derived two algorithms.
result L1-TUCKER2 outperforms other methods in tensor approximation for outlier-corrupted data.
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.
Four algorithms improve sparse tensor BR1Approx with theoretical guarantees.
problem Sparse tensor best rank-1 approximation.
method Four approximation algorithms exploiting multilinearity and sparsity.
result Theoretical worst-case approximation lower bounds for all algorithms.
DeepTD learns CNN weights from non-overlapping patches using tensor decomposition.
problem Learning weights of a deep convolutional neural network (CNN).
method Deep Tensor Decomposition (DeepTD) based on rank-1 tensor decomposition.
result DeepTD is data-efficient and works as soon as sample size exceeds total number of weights.
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…
A tensor-based model reduces weight parameters and spatial structure for high-order data classification.
problem High-order data classification with limited training samples and preserved spatial structure.
method Rank-1 FNN model based on modified feedforward neural network with rank-1 canonical decomposition and new learning algorithm.
result The proposed model outperforms state-of-the-art methods, especially in cases with small training samples.
GETF efficiently decomposes large-scale Boolean tensors.
problem Efficiently factorizing large-scale Boolean tensors.
method Geometric Expansion for all-order Tensor Factorization (GETF).
result GETF significantly improves reconstruction accuracy and efficiency.
Researchers find optimal dictionaries for minimizing average squared coefficients in random vector representations.
problem Finding optimal dictionaries for minimizing the average squared coefficients in random vector representations.
method Using rank-1 decompositions and majorization theory, the study provides a complete characterization of optimal dictionaries.
result Complete characterization of ℓ2-optimal dictionaries with polynomial time algorithms. 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…
TSL learns separable models to avoid signal cancellation and off-support extrapolation.
problem Signal cancellation and off-support extrapolation in additive models.
method Tensor Separation Learning (TSL) via stagewise greedy procedure with orthogonal refitting.
result TSL avoids information loss caused by marginalizing higher-order interactions.
New findings on tensor decomposition complexity, showing polynomial functions can estimate the largest component under certain conditions.
problem The complexity of tensor decomposition, especially for low-degree polynomials.
method Modeling a slightly larger component in a random tensor decomposition and using polynomial functions to estimate it.
result Polynomial functions can accurately estimate the largest component when r≪n3/2 but fail when r≫n3/2. New compression technique reduces RNN size by 2-4x without sacrificing accuracy.
problem Large and compute-intensive RNNs on edge devices with run-time constraints.
method Hybrid Matrix Decomposition (HMD) splits weight matrix into unconstrained and rank-1 blocks.
result HMD achieves 2-4x compression with faster run-time and similar accuracy.
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…
Proves conditions for Fourier transforms in rank 1 symmetric spaces.
problem Understanding Fourier transform bounds in symmetric spaces.
method Proves sufficient and necessary conditions using Lipschitz and Fourier type integral conditions.
result Establishes bounds for Fourier transforms in rank 1 symmetric spaces with specific moduli of continuity.
Study shows dynamics of rank 1 orbifolds in flat surfaces.
problem Characterize dynamics of rank 1 affine invariant orbifolds.
method Analyzes M-isoperiodic foliations and their ergodic properties.
result Leaves of the isoperiodic foliation are either all closed or all dense.
Sharp isoperimetric inequalities for Neumann eigenvalues in symmetric spaces.
problem Finding bounds for eigenvalues of Neumann Laplacian on domains in symmetric spaces.
method Proving sharp inequalities for eigenvalues in compact and noncompact rank-1 symmetric spaces.
result Generalization of previous results for hyperbolic space and symmetric spaces.
Right inverse found for Cartan differential in rank-1 symmetric spaces.
problem Finding a right inverse for the Cartan differential in symmetric spaces.
method Integral operator approach to the Cartan differential on exact forms.
result Extension of Gauss linking integral to rank-1 symmetric spaces.
A1GM method improves efficiency in reconstructing missing data using KL divergence.
problem Efficiently reconstructing missing data in matrices.
method Fast non-gradient-based rank-1 NMF using KL divergence.
result A1GM outperforms gradient methods in efficiency with competitive reconstruction errors.
The paper describes hyperkähler geometry of cotangent bundles using rank-1 projections.
problem Understanding hyperkähler geometry of cotangent bundles via algebraic methods.
method Algebraic description via the scheme of rank-1 projections, isometric embeddings, and generalizations.
result Explicit isometric embeddings and generalizations of hyperkähler geometry.
Joint blind source separation (J-BSS) is an emerging data-driven technique for multi-set data-fusion. In this paper, J-BSS is addressed from a tensorial perspective. We show how, by using second-order multi-set statistics in J-BSS, a specific double coupled canonical polyadic decomposition (DC-CPD) problem can be formu…
Volume comparison theorem for rank 1 symmetric spaces proved.
problem Volume comparison for symmetric spaces of non-compact type.
method Normalized Ricci--DeTurck flow to analyze volume functional and derive monotonicity properties.
result Volume comparison theorem established for rank 1 symmetric spaces of non-compact type.
Paper presents a rank-1 approximation method for natural policy gradients in deep RL.
problem Computing natural gradients requires inverting the Fisher Information Matrix, which is computationally expensive.
method Develops a rank-1 approximation to the inverse Fisher Information Matrix for efficient natural policy optimization.
result The rank-1 approximation converges faster and has similar sample complexity to stochastic policy gradient methods.
Research examines rank 1 abelian subgroups in 2-knot groups.
problem Identifying rank 1 abelian normal subgroups in 2-knot groups.
method Analyzes properties of 2-knot groups and their subgroups.
result Either 2-knot groups have no minimal Seifert hypersurface or they are topologically equivalent to a specific example.
Defines foliation criterion for dense isoperiodic leaves in rank 1 affine orbifolds.
problem Dynamics of isoperiodic leaves in rank 1 affine invariant suborbifolds.
method Defines foliation FM and establishes density criterion.
result Establishes criterion for density of isoperiodic leaves.
Rank-1 BNNs improve efficiency and scalability of Bayesian neural nets.
problem Underfitting and lack of scalability in Bayesian neural networks.
method Propose a rank-1 parameterization of BNNs and use mixture approximate posteriors.
result Rank-1 BNNs achieve state-of-the-art performance across various datasets.
Study on diagonal and separating coordinates for symmetric spaces of rank 1.
problem Existence and nonexistence of diagonal and separating coordinates for symmetric spaces of rank 1.
method Generalization of results by Gauduchon and Moroianu, 2020, and analysis of constant sectional curvature and orthogonal separation of variables.
result Diagonal coordinates exist if and only if the symmetric space has constant sectional curvature.
Dictionaries are collections of vectors used for representations of random vectors in Euclidean spaces. Recent research on optimal dictionaries is focused on constructing dictionaries that offer sparse representations, i.e., ℓ0-optimal representations. Here we consider the problem of finding optimal dictionaries …
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.
STARK learns structured dictionaries for tensor data.
problem Representing multidimensional data with structured dictionaries.
method Solves a convex relaxation of a nonconvex rank-1 tensor recovery problem.
result Empirical results show promising performance for tensors of any order.
The article proves a unique invariant measure for geodesic flows on certain rank 1 manifolds.
problem Existence and uniqueness of invariant measure for geodesic flows.
method Using Patterson-Sullivan measure and Busemann density.
result Geodesic flow on compact rank 1 manifolds has a unique invariant measure of maximal entropy.
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…
The study classifies Hessian rank 1 hypersurfaces in dimensions 2, 3, and 4.
problem Classifying Hessian rank 1 affinely homogeneous hypersurfaces in specific dimensions.
method Power Series Method of Equivalence, infinitesimal calculations.
result Identified all non-product constant Hessian rank 1 affinely homogeneous hypersurfaces in dimensions 2, 3, and 4.
Improved sample and time complexity for identifying mixtures of product distributions.
problem Identifying a mixture of k product distributions from statistics. method Combining robust tensor decomposition and Hadamard extensions to bound the condition number of key matrices.
result Achieved sample complexity and run-time complexity of (1/ζ)O(k) for n≥2k−1. A new algorithm completes rank-1 tensors with minimal samples and time.
problem Completing rank-1 tensors with minimal samples and time.
method Gauss-Jordan on random linear systems.
result Gauss-Jordan algorithm uses O(d2logd) samples and runs in O(md2) time. Paper proves geodesic ball maximizes second Robin eigenvalue in non-compact symmetric spaces.
problem Maximizing the second Robin eigenvalue in non-compact rank-1 symmetric spaces.
method Quantitative spectral inequality for the second Robin eigenvalue.
result Geodesic ball maximizes the second Robin eigenvalue among domains of the same volume.
We discuss the Morse-Novikov cohomology of a compact manifold, associated to a closed one--form whose free abelian group generated by its periods ⟨∫γη∣[γ]∈π1(M)⟩ is of rank 1, the focus being on locally conformally symplectic manifolds. In particular, we provide an explicit computation for t…
We define and study the renormalized volume for geometrically finite hyperbolic 3-manifolds, including with rank-1 cusps. We prove a variation formula, and show that for certain families of convex co-compact hyperbolic metrics $g_\eps$ degenerating to a geometrically finite hyperbolic metric g0 with rank-1 cus…
New compact ECS manifolds with rank 2 discovered, differing from previous rank 1 examples.
problem Finding new compact ECS manifolds with rank 2.
method Constructing new examples of compact pseudo-Riemannian manifolds with parallel Weyl tensor, rank 1 or 2.
result New compact ECS manifolds of rank 2, locally homogeneous, and geodesically incomplete.
We analyze the structure of covariance matrices under graph constraints.
problem Analyzing the structure of covariance matrices under graph constraints.
method We explore the algebraic structure of the solution space of convex optimization problem Constrained Minimum Trace Factor Analysis (CMTFA) under a latent star topology.
result CMTFA can have either a rank 1 or a rank n-1 solution, with conditions for both.
Data compression speeds up machine learning loss calculations.
problem Computational demand in calculating mean squared error for large datasets.
method Use rank-1 lattices to compress data, assigning weights based on original data and responses.
result Our QMC data compression algorithms can lead to arbitrary high convergence rates for smooth functions.
Let M be a closed hypersurface in a simply connected rank-1 symmetric space $\olm$. In this paper, we give an upper bound for the first eigenvalue of the Laplacian of M in terms of the Ricci curvature of $\olm$ and the square of the length of the second fundamental form of the geodesic spheres with center at the ce…
Gradient Descent with small random initialization solves rank-1 matrix completion efficiently.
problem Matrix completion for rank-1 symmetric matrices.
method Gradient Descent with small random initialization.
result Gradient Descent converges to the ground truth for rank-1 symmetric matrix completion.
This work shows that a simple local search can recover true principal components in non-negative rank-1 RPCA.
problem Recovering true principal components in non-negative rank-1 robust principal component analysis with noisy measurements.
method Using the Burer-Monteiro approach to cast RPCA as a non-convex and non-smooth ℓ1 optimization problem. result The low-dimensional formulation of symmetric and asymmetric positive rank-1 RPCA has a unique global solution and no spurious local solutions.
In this paper, we establish that, for statistically convex-cocompact actions, contracting elements are exponentially generic in counting measure. Among others, the following exponential genericity results are obtained as corollaries for the set of hyperbolic elements in relatively hyperbolic groups, the set of rank-1 e…
It has long been conjectured that hypotheses spaces suitable for data that is compositional in nature, such as text or images, may be more efficiently represented with deep hierarchical networks than with shallow ones. Despite the vast empirical evidence supporting this belief, theoretical justifications to date are li…