The main results of our paper deal with the lifting problem for multilinear differential operators between complexes of horizontal de Rham forms on the infinite jet bundle. We answer the question when does an n-multilinear differential operator from the space of (N,0)-forms (where N is the dimension of the base) to the…
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48 results for “multilinear map”
Derives a primal-dual MLSVD formulation for multilinear data.
problem Efficiently decompose multilinear data for signal analysis and deep learning.
method Kernelizable primal-dual formulation of MLSVD.
result Derives a new MLSVD formulation with computational advantages.
Proposes MLDP for modeling multilinear data.
problem Handling data with interactions from multiple factors.
method Combines Dirichlet processes with multilinear factor analysis.
result Achieved state-of-the-art performance on real-world data.
The paper extends Minkowski's theorem to a broader class of Finsler manifolds.
problem Extending Minkowski's second theorem to more general Finsler manifolds.
method Analyzing the non-vanishing condition for the hyperdeterminant reduced modulo 2 of the multilinear map induced by the fundamental class.
result The principle of Minkowski's second theorem holds for a larger class of Finsler manifolds.
New method forecasts multilinear data using tensor autoregression.
problem Forecasting 2D data in big data.
method L-Transform Tensor autoregressive (L-TAR) method.
result Statistical independence achieved through invertible discrete linear transforms.
Geometrically, tensors of fixed rank form a minimal submanifold.
problem Understanding the geometric properties of tensors of fixed rank.
method Geometric analysis of tensors in Euclidean space.
result Real tensors of fixed multilinear rank form a minimal submanifold.
Analytic Lie rack structures on Leibniz algebras are characterized and rigid Lie algebras are identified.
problem Characterizing and identifying rigid Lie algebras with analytic Lie rack structures.
method Analytic Lie rack structures are defined and characterized using multilinear equations and cohomological interpretations.
result Simple Lie algebras are conjectured to be rigid as left Leibniz algebras.
Unified multilinear model for causal factor disentanglement.
problem Disentangling causal factors from complex data without direct manipulation.
method Hierarchical block multilinear factorization (M-mode Block SVD) and incremental approach.
result Interpretable object representation robust to occlusion and reduced training data.
New algorithm solves -norm constrained multilinear logistic regression for tensor data.
problem Non-convex and nonsmooth -norm constraints in multilinear logistic regression.
method APALM method for globally convergent optimization.
result APALM ensures convergence to a first-order critical point.
Tucker decomposition is the cornerstone of modern machine learning on tensorial data analysis, which have attracted considerable attention for multiway feature extraction, compressive sensing, and tensor completion. The most challenging problem is related to determination of model complexity (i.e., multilinear rank), e…
The goal of tensor completion is to fill in missing entries of a partially known tensor (possibly including some noise) under a low-rank constraint. This may be formulated as a least-squares problem. The set of tensors of a given multilinear rank is known to admit a Riemannian manifold structure, thus methods of Rieman…
In this paper we present a new model and an algorithm for unsupervised clustering of 2-D data such as images. We assume that the data comes from a union of multilinear subspaces (UOMS) model, which is a specific structured case of the much studied union of subspaces (UOS) model. For segmentation under this model, we de…
Wide neural networks become linear, but adding bottlenecks makes them bilinear or multilinear.
problem Understanding the transition of neural networks from linearity to higher-order functions.
method Analyzing the behavior of randomly initialized wide neural networks with and without bottleneck layers.
result Bottleneck layers transform the network's function from linear to bilinear or multilinear.
GMT improves interpretability of XGNNs by approximating SubMT.
problem Limited understanding of existing interpretable subgraph learning methods.
method Formulated subgraph multilinear extension (SubMT) and designed GMT architecture.
result GMT outperforms state-of-the-art in both interpretability and generalizability.
Study uses random matrix theory to improve tensor approximation accuracy.
problem Improving tensor approximation accuracy in the presence of noise.
method Random matrix theory applied to tensor unfoldings.
result Characterizes spectral behavior of tensor unfoldings and predicts reconstruction performance.
Principal component analysis (PCA) is an unsupervised method for learning low-dimensional features with orthogonal projections. Multilinear PCA methods extend PCA to deal with multidimensional data (tensors) directly via tensor-to-tensor projection or tensor-to-vector projection (TVP). However, under the TVP setting, i…
Study of differential forms and vector fields on orbit spaces.
problem Understanding vector fields and differential forms on orbit spaces.
method Defined differential forms and vector fields as multilinear maps on infinitesimal diffeomorphisms.
result Intrinsic view of vector fields and differential forms on orbit spaces.
Matrix factorizations and their extensions to tensor factorizations and decompositions have become prominent techniques for linear and multilinear blind source separation (BSS), especially multiway Independent Component Analysis (ICA), NonnegativeMatrix and Tensor Factorization (NMF/NTF), Smooth Component Analysis (Smo…
Proposes FMPCA for federated tensor data dimensionality reduction.
problem Integration of MPCA into federated learning.
method Federated Multilinear Principal Component Analysis (FMPCA).
result FMPCA preserves performance of traditional MPCA in federated learning.
In this paper, we study a nonconvex continuous relaxation of MAP inference in discrete Markov random fields (MRFs). We show that for arbitrary MRFs, this relaxation is tight, and a discrete stationary point of it can be easily reached by a simple block coordinate descent algorithm. In addition, we study the resolution …
The paper defines supermanifolds via multilinear bundles and shows their structure.
problem Defining and understanding supermanifolds in a clear, accessible way.
method Categorical approach, using multilinear bundles and projective limits.
result Supermanifolds can be seen as infinite-dimensional fiber bundles.
Given a multifunction from to the fold symmetric product , we use the Dold-Thom Theorem to establish a homological selection Theorem. This is used to establish existence of Nash equilibria. Cost functions in problems concerning the existence of Nash Equilibria are traditionally multilinear in the mixe…
New model generates unseen attribute combinations from limited data.
problem Lack of generalization in deep generative models for unseen attribute combinations.
method Introduces multilinear latent conditioning to capture multiplicative interactions.
result Demonstrates effectiveness on MNIST, Fashion-MNIST, and CelebA datasets.
Dimensionality reduction is a main step in the learning process which plays an essential role in many applications. The most popular methods in this field like SVD, PCA, and LDA, only can be applied to data with vector format. This means that for higher order data like matrices or more generally tensors, data should be…
MCCA extracts shared structure from multiple tensor datasets.
problem Extracting shared structure from multiple tensor datasets.
method Multilinear common component analysis (MCCA) using Kronecker products of mode-wise covariance matrices.
result MCCA constructs a common basis that retains information from multiple tensor datasets.
Efficiently optimizes boolean functions using multilinear polynomials and exponential weight updates.
problem Optimizing boolean functions over the boolean hypercube with high computational cost.
method Proposes a computationally efficient algorithm using multilinear polynomials and exponential weight updates.
result Improves computational time up to several orders of magnitude compared to state-of-the-art algorithms.
We study algebraic varieties of ReLU networks to understand their representable functions.
problem Understanding the functions that ReLU neural networks can represent.
method We introduce algebraic varieties associated with ReLU networks and derive polynomial equations to characterize representable functions.
result Conditions under which ReLU networks attain their expected dimension, providing insight into their structural properties.
The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed in recent work (collaboration with H. Gloeckner and K.-H. Neeb), without any restriction on the dimension or on the cha…
Paper introduces a new multilinear functional for spectral triples and computes its properties.
problem Computing properties of spectral triples and their associated Hodge operators.
method Introduces a new multilinear functional for spectral triples and computes its properties using noncommutative residue and perturbed de-Rham Hodge operators.
result Recover two forms, torsion of the linear connection, and four forms by the noncommutative residue and perturbed de-Rham Hodge Dirac triple.
Causal Deep Learningcs.LG
Causal deep learning tackles causal inference using tensor factor analysis.
problem Addressing causal questions in data using neural networks.
method Tensor factor analysis and neural network architectures (causal capsules, tensor transformer, multilinear projection algorithm).
result Derives deep neural networks for causal inference with tensor factor analysis.
Extends De Leeuw theorems to noncommutative groups and multipliers.
problem Proving bounds for Fourier multipliers on noncommutative groups.
method Analyzing Fourier multipliers on discrete subgroups of locally compact groups.
result Established bounds for Fourier multipliers on noncommutative groups.
Nowadays, with the availability of massive amount of trade data collected, the dynamics of the financial markets pose both a challenge and an opportunity for high frequency traders. In order to take advantage of the rapid, subtle movement of assets in High Frequency Trading (HFT), an automatic algorithm to analyze and …
Several Approximation Algorithms for Sparse Best Rank-1 Approximation to Higher-Order Tensorsmath.NA
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.
A new MLDS model captures nonlinear tensor time series with improved accuracy and efficiency.
problem Modeling and analyzing nonlinear tensor time series data.
method Transform-based multilinear dynamical system (MLDS) with EM algorithm for parameter estimation.
result Significantly higher prediction accuracy and exponential improvement in training time compared to state-of-the-art models.
We give an algorithm for completing an order- symmetric low-rank tensor from its multilinear entries in time roughly proportional to the number of tensor entries. We apply our tensor completion algorithm to the problem of learning mixtures of product distributions over the hypercube, obtaining new algorithmic result…
Algorithm identifies sources in product distributions with improved complexity.
problem Identifying sources in mixtures of product distributions.
method Approximate multilinear moments input, 2^{O(k^2)} n^{O(k)} operations.
result First explicit bound on computational complexity of source identification.
We tackle binary tensor decomposition with a multilinear model and likelihood-based estimation.
problem Decomposing binary tensors with probabilistic models.
method Multilinear Bernoulli model, rank-constrained likelihood estimation, alternating optimization.
result The estimation error bound is established and shown to be minimax optimal.
We are interested in approximation of a multivariate function by linear combinations of products of univariate functions , . In the case it is a classical problem of bilinear approximation. In the case of approximation in the space the bili…
Extends RRR to capture nonlinear interactions in multi-response regression.
problem Complex relationships in real-world data cannot be adequately modeled by linear interactions.
method Introduces Higher Order Reduced Rank Regression (HORRR) using tensor representations and Tucker decomposition.
result HORRR can capture nonlinear interactions in multi-response regression.
We study conformal deformation problems on manifolds with boundary which include prescribing in the interior. In particular, we prove a Dirichlet principle when the induced metric on the boundary is fixed and an Obata-type theorem on the upper hemisphere. We introduce some conformally covariant multilinear…
Tensor PCA for Factor Modelsecon.EM
Optimal tensor PCA for estimating factors and loadings in high-dimensional panel data.
problem Estimating factors and loadings in high-dimensional panel data with non-negligible correlations.
method Tensor Principal Component Analysis (TPCA) for estimating factors and loadings in a tensor factor model.
result Simple TPCA is optimal for strong factors and can be improved for weak factors with alternating least-squares iterations.
We introduce the problem of learning mixtures of subcubes over , which contains many classic learning theory problems as a special case (and is itself a special case of others). We give a surprising -time learning algorithm based on higher-order multilinear moments. It is not possible to l…
We propose a new framework for the analysis of low-rank tensors which lies at the intersection of spectral graph theory and signal processing. As a first step, we present a new graph based low-rank decomposition which approximates the classical low-rank SVD for matrices and multi-linear SVD for tensors. Then, building …
Nonnegative Tucker decomposition (NTD) is a powerful tool for the extraction of nonnegative parts-based and physically meaningful latent components from high-dimensional tensor data while preserving the natural multilinear structure of data. However, as the data tensor often has multiple modes and is large-scale, exist…
The paper tackles high-dimensional function approximation using tree-based tensor formats.
problem Approximating high-dimensional functions in statistical learning.
method Empirical risk minimization over tree-based tensor formats, exploiting multilinear models and sparsity.
result Numerical stability and reliability of the proposed algorithms for learning.
Rank-R FNN handles high-dimensional data efficiently.
problem Handling irregularities in high-dimensional data.
method Imposes Canonical/Polyadic decomposition on parameters.
result Achieves state-of-the-art performance on higher-order tensor data.
New sampling method estimates Shapley values more accurately.
problem Exponential time complexity of computing Shapley values.
method Multilinear sampling algorithm based on game theory.
result Our method reduces variance and provides more accurate Shapley value estimations.
New method accelerates CNNs for mobile devices by approximating tensors and quantizing weights.
problem Efficiently compress and accelerate CNNs for mobile devices.
method Low-rank tensor approximation in Tucker format combined with quantization of weights and activations.
result Our method significantly improves CNN performance on various classification tasks.