Paper characterizes optimization landscape of Tucker decomposition.
problem Finding exact Tucker decomposition is a nonconvex optimization problem.
method Characterized the optimization landscape and provided a local search algorithm.
result All local minima are globally optimal if tensor has an exact Tucker decomposition.
Short proof for ideal polygons with near optimal orthogeodesic decomposition.
problem Decomposing ideal polygons into orthogeodesics.
method Short proof with orthogeodesic decomposition of length at most 2log(n). result Optimal orthogeodesic decomposition of ideal polygons with length 2log(n). Unified algorithm for tensor decomposition supports multiple loss functions and models.
problem Efficient tensor decomposition for various models and loss functions.
method Hierarchical combination of ADMM and MM for optimization.
result Wide-range applications can be solved by the proposed algorithm.
Efficient algorithm for Hadamard decomposition of matrices.
problem Decomposing matrices into low-rank factors efficiently.
method Alternating optimization with SVD-inspired initialization and momentum.
result Significantly improved performance compared to existing methods.
Revisits CP tensor decomposition for noisy, non-orthogonal data.
problem Statistical optimality and convergence of ALS in noisy, non-orthogonal, higher-rank settings.
method Statistical analysis and TASD method for initialization.
result ALS with TASD achieves optimal error in rank-one setting within one or two iterations.
MARS automatically selects tensor decomposition ranks, improving performance in neural network tasks.
problem Determining optimal decomposition ranks in tensor decompositions.
method MARS uses binary masks to learn optimal tensor structure during training via relaxed MAP estimation.
result MARS achieves better results than previous methods in various tasks.
New method uses random decompositions for high-dimensional Bayesian optimization.
problem Learning accurate decompositions for high-dimensional black-box functions.
method Data-independent random tree-based decomposition sampling.
result Random decomposition upper-confidence bound algorithm (RDUCB) yields significant empirical gains.
This work studies the linear approximation of high-dimensional dynamical systems using low-rank dynamic mode decomposition (DMD). Searching this approximation in a data-driven approach is formalised as attempting to solve a low-rank constrained optimisation problem. This problem is non-convex and state-of-the-art algor…
Optimal Euclidean structure minimizes energy in weighted toroidal graphs.
problem Finding the optimal Euclidean structure for weighted toroidal graphs.
method Minimizing Dirichlet energy over all possible Euclidean structures and realizations within a fixed homotopy class.
result The optimal Euclidean structure induces a weighted Delaunay decomposition.
New method solves KP problem using global Cartan decompositions.
problem Solving time-optimal unitaries for targets in semi-simple Lie groups.
method Global Cartan decompositions of symmetric spaces for optimal control.
result Analytical solutions for time-optimal unitaries under specific conditions.
APINNs improve physics-informed neural networks through flexible domain decomposition.
problem Improving physics-informed neural networks (PINNs) for solving partial differential equations (PDEs).
method Introduces a trainable gate network for soft domain decomposition, allowing flexible parameter sharing and improved generalization.
result APINNs significantly improve PINNs and XPINNs, demonstrating better performance on various types of PDEs.
Optimal discrete harmonic maps between hyperbolic surfaces are found via minimizing energy.
problem Finding optimal discrete harmonic maps between hyperbolic surfaces.
method Minimizing Dirichlet energy over all possible hyperbolic structures and realizations within a fixed homotopy class.
result At the optimal hyperbolic structure, the discrete harmonic map and edge weights are induced from a weighted Delaunay decomposition.
Tensor decomposition, a collection of factorization techniques for multidimensional arrays, are among the most general and powerful tools for scientific analysis. However, because of their increasing size, today's data sets require more complex tensor decomposition involving factorization with multiple matrices and dia…
Study uncovers complex critical points in tensor decomposition.
problem Nonconvex optimization of symmetric tensor decomposition.
method Utilized symmetry to construct critical points and analyze Hessian.
result Obtained precise analytic estimates on objective function and Hessian.
Study optimal investment with herd behavior using rational decision decomposition.
problem Optimal investment problem considering herd behavior between two agents.
method Introduce average deviation term, use variational method, rational decision decomposition, investment opinion.
result Quantitative analysis of herd behavior impact on investment decisions.
IKD uses eigen-decomposition for nonlinear dimensionality reduction.
problem Lack of sophisticated and nonlinear dimensionality reduction methods.
method Inverse Kernel Decomposition (IKD) based on eigen-decomposition of sample covariance matrix.
result IKD achieves comparable performance to optimization-based methods with faster running speeds.
Paper tackles tensor decomposition for unaligned observations using RKHS and novel loss functions.
problem Tackles tensor decomposition for unaligned observations.
method Uses functions in RKHS to represent mode with unaligned observations, introduces versatile loss function, proposes optimization algorithm and stochastic gradient method.
result Demonstrates improved tensor decomposition efficiency and effectiveness with synthetic and real data.
Paper proves optimal decomposition for matrix fields, reducing convex integration steps.
problem Optimizing decomposition of symmetric matrix fields for convex integration.
method Algebraic geometry and topology applications to prove optimality.
result Optimal decomposition with fewer rank-one terms, improving Hölder regularity.
Optimal Reeb graphs identified for polygon decomposition.
problem Investigating the topological structure of planar polygon decomposition.
method Using oriented Reeb graphs with a marked vertex for height functions.
result Described all possible optimal Reeb graphs for specific polygon configurations.
NOHD optimizes multi-agent systems by decomposing dynamics into irrotational and solenoidal components.
problem Non-stationarity and conflicting interests in multi-agent learning problems.
method NOHD (Newton Optimization on Helmholtz Decomposition) decomposes system dynamics into irrotational and solenoidal components.
result NOHD ensures quadratic convergence in purely irrotational and solenoidal systems and attracts to stable fixed points in general multi-agent systems.
Researchers decompose harmonic forms on specific types of manifolds.
problem Decomposing harmonic forms on compact almost-Kähler manifolds.
method Proved primitive decompositions of Dolbeault harmonic forms in specific bidegrees.
result Primitive decompositions of ∂-, ∂-harmonic forms in bidegree (1,1) and (n−1,n−1). Tensor decomposition methods are widely used for model compression and fast inference in convolutional neural networks (CNNs). Although many decompositions are conceivable, only CP decomposition and a few others have been applied in practice, and no extensive comparisons have been made between available methods. Previo…
RieCUR improves Robust PCA by combining Riemannian optimization and CUR decompositions.
problem Robust Principal Component Analysis (PCA) to recover low-rank and sparse matrices from their sum.
method Riemannian CUR (RieCUR) algorithm that combines Riemannian optimization and robust CUR decompositions.
result RieCUR achieves state-of-the-art performance in Robust PCA with improved robustness to outliers and comparable computational complexity.
Develops SymGCP for tensor decompositions with general symmetry.
problem Handling symmetry in tensor decompositions for better model accuracy.
method Introduces SymGCP, a generalized CP decomposition that accounts for any subset of tensor modes' symmetry.
result SymGCP enables efficient and scalable tensor decomposition with improved model robustness and accuracy.
Study primitive decompositions for harmonic forms on almost Kähler manifolds.
problem Decomposing harmonic forms on almost Kähler manifolds.
method Proved primitive decompositions for Bott-Chern and Aeppli harmonic forms in specific bidegrees.
result Optimal bidegrees for primitive decompositions of harmonic forms.
In tensor completion tasks, the traditional low-rank tensor decomposition models suffer from the laborious model selection problem due to their high model sensitivity. In particular, for tensor ring (TR) decomposition, the number of model possibilities grows exponentially with the tensor order, which makes it rather ch…
An efficient algorithm optimizes trades across CFMM networks.
problem Optimizing trades through a network of CFMMs for maximum utility.
method Decomposition method to solve the routing problem.
result Significant performance improvements over commercial solvers.
Optimized DMD for fast atmospheric chemistry forecasting.
problem Forecasting global atmospheric chemistry dynamics efficiently.
method Optimized Dynamic Mode Decomposition (DMD) for reduced order modeling.
result Significant improvement in computational speed and interpretability.
We analyze stochastic gradient descent for optimizing non-convex functions. In many cases for non-convex functions the goal is to find a reasonable local minimum, and the main concern is that gradient updates are trapped in saddle points. In this paper we identify strict saddle property for non-convex problem that allo…
New algorithm improves dynamic mode decomposition for high-dimensional data.
problem Reduced modeling in high-dimensional spaces.
method Low rank constraint optimization and kernel-based computation.
result Gain in approximation accuracy and computational efficiency.
Gradient descent can find better tensor decompositions than lazy training in over-parameterized settings.
problem Finding better tensor decompositions in over-parameterized settings.
method Gradient descent on over-parameterized tensor decomposition problems.
result Gradient descent can find an approximate tensor decomposition with rank m=O∗(r2.5llogd), while lazy training requires m=Ω(dl−1). Dynamic Mode Decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of non-linear systems from experimental datasets. Recently, several attempts have extended DMD to the context of low-rank approximations. This extension is of particular interest for reduced-order modeling in various applicative …
Physics-inspired methods optimize SVD compression of LLMs.
problem Efficiently compressing large language models (LLMs) using SVD.
method FermiGrad for globally optimal rank selection and PivGa for lossless compression.
result Global optimization of SVD ranks and lossless compression of low-rank factors.
In this paper we de ne conditional random elds in reproducing kernel Hilbert spaces and show connections to Gaussian Process classi cation. More speci cally, we prove decomposition results for undirected graphical models and we give constructions for kernels. Finally we present e cient means of solving the optimization…
The space of probability densities is an infinite-dimensional Riemannian manifold, with Riemannian metrics in two flavors: Wasserstein and Fisher--Rao. The former is pivotal in optimal mass transport (OMT), whereas the latter occurs in information geometry---the differential geometric approach to statistics. The Rieman…
Continuous optimization is an important problem in many areas of AI, including vision, robotics, probabilistic inference, and machine learning. Unfortunately, most real-world optimization problems are nonconvex, causing standard convex techniques to find only local optima, even with extensions like random restarts and …
We present an approach for penalized tensor decomposition (PTD) that estimates smoothly varying latent factors in multi-way data. This generalizes existing work on sparse tensor decomposition and penalized matrix decompositions, in a manner parallel to the generalized lasso for regression and smoothing problems. Our ap…
Singular Value Decomposition (SVD) constitutes a bridge between the linear algebra concepts and multi-layer neural networks---it is their linear analogy. Besides of this insight, it can be used as a good initial guess for the network parameters, leading to substantially better optimization results.
We consider the problem of decomposing a multivariate polynomial as the difference of two convex polynomials. We introduce algebraic techniques which reduce this task to linear, second order cone, and semidefinite programming. This allows us to optimize over subsets of valid difference of convex decompositions (dcds) a…
New bounds found for optimizing non-convex functions with noisy data.
problem Limits of first-order stochastic optimization in non-convex settings.
method Divergence decomposition to construct challenging subclasses.
result Sharp lower bounds on noisy gradient queries for various non-convex classes.
This work improves tensor decomposition methods, especially for large datasets.
problem Lack of efficient methods for estimating Tucker decompositions.
method Applies Johnson-Lindenstrauss type guarantees to Tucker decompositions with random embeddings.
result Effective dimension reduction with minimal error for large tensors.
Optimizes mixture models without parametrizing distributions using tensor decomposition.
problem Estimating conditionally-independent mixture models in high dimensions.
method Alternating least squares optimization scheme for tensor decomposition.
result Competitive performance and applicability to various models and applications.
Paper proposes a new optimization framework for learning eigenfunctions of operators.
problem Computing eigenvalue decomposition of high-dimensional operators.
method Operator SVD with Neural Networks via Nested Low-Rank Approximation.
result Proposed method efficiently learns top-L singular values and functions in the correct order.
The paper tackles fair correlation clustering with new algorithms and analysis.
problem Fair variants of correlation clustering under various constraints.
method Introducing a novel combinatorial optimization problem for fairlet decomposition.
result Approximation algorithms for fair correlation clustering under multiple fairness constraints.
We analyze bias-variance of margin losses.
problem Understanding model overfitting/underfitting.
method Bias-variance decomposition for strictly convex margin losses.
result Expected risk decomposes into central model risk and data variation.
Paper unifies subspace identification and DMD for dynamical systems.
problem Estimating dynamical models from data.
method Unified optimization and regression problems for SID and DMD.
result Proves equivalence of SID and DMD for optimal model construction.
New recommendations improve Gaussian process accuracy and stability.
problem Numerical instabilities and poor test likelihoods in iterative Gaussian process learning.
method Investigated CG tolerance, preconditioner rank, and Lanczos decomposition rank. Recommended small CG tolerance and large root decomposition size.
result L-BFGS-B optimizer achieves convergence with fewer gradient updates, improving Gaussian process accuracy.
Study on efficient estimation of Gaussian mean with limited communication.
problem Estimating Gaussian mean under communication constraints.
method Decomposition into localization and refinement stages, development of communication-efficient and statistically optimal procedures.
result Established minimax rates of convergence and developed optimal procedures.