New bound for neural networks with full-rank weights, independent of network width.
problem Understanding generalization of neural networks with full-rank weight matrices.
method Using Koopman operators to derive a tighter generalization bound for full-rank weight matrices.
result The bound is tighter than existing norm-based bounds when condition numbers are small.
Completes the space of vector-valued one-forms on manifolds.
problem Metric incompleteness of the space of full-ranked one-forms.
method Distance equality and quotient structures.
result Concrete description of the metric completion of the space of full-ranked one-forms.
Study Betti and Hodge numbers of solvmanifolds from integer polynomials.
problem Computing Betti and Hodge numbers of solvmanifolds constructed from integer polynomials.
method Analyzing de Rham and Dolbeault cohomology of solvmanifolds under algebraic conditions.
result Explicit generating polynomials for Hodge numbers in quasi full rank case.
New metrics defined for full-rank correlation matrices, ensuring unique operations.
problem No suitable problem statement as the abstract does not describe a problem to be solved.
method New Riemannian metrics defined on full-rank correlation matrices, providing unique operations.
result Unique Riemannian logarithm and Fréchet mean defined for full-rank correlation matrices.
Gradient descent recovers planted weights in shallow neural networks with quadratic activations.
problem Learning shallow neural networks with quadratic activations and planted weights.
method Analysis of optimization landscape, gradient descent, semicircle law for Wishart ensemble.
result Gradient descent can recover planted weights if initialized below an energy barrier.
New method for initializing low-rank neural networks improves performance.
problem Training low-rank neural networks efficiently and accurately.
method Inspired by function approximation, proposes a novel low-rank initialization framework.
result Demonstrates significant gap between spectral and low-rank initialization approaches.
Theoretical analysis of the error landscape of deep neural networks has garnered significant interest in recent years. In this work, we theoretically study the importance of noise in the trajectories of gradient descent towards optimal solutions in multi-layer neural networks. We show that adding noise (in different wa…
This paper describes a versatile method that accelerates multichannel source separation methods based on full-rank spatial modeling. A popular approach to multichannel source separation is to integrate a spatial model with a source model for estimating the spatial covariance matrices (SCMs) and power spectral densities…
Study of Eisenstein series linked to hyperbolic cusps.
problem Understanding Eisenstein series associated with hyperbolic cusps.
method Analyzing cohomology classes and intertwining operators.
result Different cusps correspond to linearly independent cohomology classes.
We consider the problem of statistical inference for ranking data, specifically rank aggregation, under the assumption that samples are incomplete in the sense of not comprising all choice alternatives. In contrast to most existing methods, we explicitly model the process of turning a full ranking into an incomplete on…
The paper introduces structured variational families to improve scalability in black-box variational inference.
problem Scalability issues in black-box variational inference, especially for large datasets and hierarchical models.
method Developed structured variational families that achieve better iteration complexity of O(N) compared to full-rank families.
result Structured variational families can achieve better scaling with respect to dataset size N, improving iteration complexity from O(N^2) to O(N).
Optimizes wide low-rank neural networks for reduced parameters and cost.
problem Reducing the number of learnable parameters in wide neural networks.
method Analyzed edge-of-chaos dynamics and derived formulae for optimal weight and bias variances.
result Optimal weight and bias variances for low-rank networks follow from multiplicative scaling.
Paper solves Christoffel-Minkowski and Weingarten curvature problems in hyperbolic space.
problem Christoffel-Minkowski and Weingarten curvature problems in hyperbolic space.
method Proved existence of solutions using a new full rank theorem.
result Existence of smooth, origin-symmetric, strictly horospherically convex solutions.
An isomorphism of symplectically tame smooth pseudocomplex structures on the complex projective plane which is a homeomorphism and differentiable of full rank at two points is smooth.
Method quantifies uncertainty in full ranking with new CP approach.
problem Uncertainty quantification in full ranking with unknown ground truth.
method Transductive Conformal Prediction (CP) method to construct distribution-free bounds.
result Valid prediction sets and false coverage proportion control for full ranking.
This article addresses the modeling of reverberant recording environments in the context of under-determined convolutive blind source separation. We model the contribution of each source to all mixture channels in the time-frequency domain as a zero-mean Gaussian random variable whose covariance encodes the spatial cha…
The paper analyzes how low-rank layers in neural networks improve generalization.
problem Understanding how low-rank layers affect generalization in neural networks.
method Applying Maurer's chain rule for Gaussian complexity to analyze rank and spectral norm constraints.
result Deep networks with low-rank layers achieve better generalization than those with full-rank layers.
PLD distills knowledge using choice-theoretic Plackett-Luce model.
problem Model compression and knowledge transfer between large and small networks.
method PLD uses a weighted list-wise ranking loss based on the Plackett-Luce model.
result PLD achieves consistent gains across diverse architectures and distillation methods.
In this paper, we develop a relative error bound for nuclear norm regularized matrix completion, with the focus on the completion of full-rank matrices. Under the assumption that the top eigenspaces of the target matrix are incoherent, we derive a relative upper bound for recovering the best low-rank approximation of t…
Slow feature analysis (SFA) is a method for extracting slowly varying features from a quickly varying multidimensional signal. An open source Matlab-implementation sfa-tk makes SFA easily useable. We show here that under certain circumstances, namely when the covariance matrix of the nonlinearly expanded data does not …
Simple algorithms identify best items or full rankings from choice-based feedback.
problem Learning to identify the best item or full ranking from choice-based feedback.
method Nested Elimination (NE) and Nested Partition (NP) algorithms.
result NE is worst-case asymptotically optimal, NP is optimal up to a constant factor.
This paper solves the Christoffel problem in hyperbolic space and its equivalent on spheres.
problem Prescribing curvatures for convex hypersurfaces in hyperbolic space.
method Proving a full rank theorem to establish the existence of solutions.
result Existence of solutions to the Christoffel problem and its equivalent Nirenberg-Kazdan-Warner problem on spheres.
New methods provide stable ranking without assumptions on data distributions.
problem Stability issues in ranking problems with noisy data.
method Developed a stability framework and two ranking operators.
result Guaranteed stability without assumptions on data distributions.
The expressive power of a Gaussian process (GP) model comes at a cost of poor scalability in the data size. To improve its scalability, this paper presents a low-rank-cum-Markov approximation (LMA) of the GP model that is novel in leveraging the dual computational advantages stemming from complementing a low-rank appro…
PLUMAGE improves large model training efficiency and stability.
problem Accelerator memory and networking constraints during large model training.
method Probabilistic Low rank Unbiased Minimum Variance Gradient Estimator (PLUMAGE) that resolves bias and variance issues.
result PLUMAGE reduces training loss by 28% on average across the GLUE benchmark.
Geodesics found in deep linear networks.
problem Finding shortest paths in deep neural networks.
method Derived ODEs and explicit solutions for geodesics.
result Horizontal straight lines are geodesics in invariant manifold.
Low-rank MPPCA improves importance sampling in high dimensions.
problem Estimating full-rank GMM covariance matrices in high dimensions is numerically unstable.
method Use MPPCA mixtures as low-rank proposals for importance sampling in high-dimensional spaces.
result Consistent gains in sample efficiency and quality of failure distribution characterization.
In this paper, we present some theoretical work to explain why simple gradient descent methods are so successful in solving non-convex optimization problems in learning large-scale neural networks (NN). After introducing a mathematical tool called canonical space, we have proved that the objective functions in learning…
Determinantal point processes (DPPs) have garnered attention as an elegant probabilistic model of set diversity. They are useful for a number of subset selection tasks, including product recommendation. DPPs are parametrized by a positive semi-definite kernel matrix. In this work we present a new method for learning th…
New insights into attention mechanisms reveal dramatic trade-offs between rank and heads.
problem Dramatic trade-offs between rank and number of heads in attention mechanisms.
method Presented a simple target function and proved theoretical limits.
result Full-rank attention is necessary for long contexts, while low-rank is sufficient for short ones.
New algorithm ranks players from partial comparisons with optimal rate.
problem Ranking players from partial pairwise comparisons.
method Divide-and-conquer approach, local MLE within groups.
result Optimal ranking algorithm with minimax rate.
Identifying components and estimating mixing weights in unlabeled finite mixtures under marginal independence.
problem Identifying components and estimating mixing weights in unlabeled finite mixtures.
method Proving structural results and extending them to observable mixtures.
result Identifying components and estimating mixing weights under marginal independence.
New model for high rank matrix completion with online and batch methods.
problem Matrix completion for high rank matrices with latent structure.
method Kernel trick to map data into a high dimensional feature space, explicit parametrization of low dimensional subspace, online fitting procedure.
result Online method can handle streaming data and adapt to non-stationary latent structure.
Analyzes Hessian spectrum for neural networks near optimal learning.
problem Understanding learning dynamics near optimal points in neural networks.
method Characterizes Hessian eigenspectrum for teacher-student problems, using analytical and numerical methods.
result The rank of the Hessian matrix determines effective number of parameters for non-linear networks.
Simple perturbation of Vafa-Witten equations leads to transversality.
problem Transversality of Vafa-Witten moduli space.
method Simple perturbation of Vafa-Witten equations, proving transversality for SU(2) or SO(3) structure groups. result For generic perturbation parameter, the full rank part of the moduli space satisfies transversality.
New metrics improve landing algorithms for orthogonality constraints.
problem Optimizing landing algorithms with orthogonality constraints.
method Proposed a family of metrics over full-rank matrices to enhance landing algorithms.
result Natural extension of β-metric improves landing performance.
High-dimensional linear classifiers, such as the support vector machine (SVM) and distance weighted discrimination (DWD), are commonly used in biomedical research to distinguish groups of subjects based on a large number of features. However, their use is limited to applications where a single vector of features is mea…
We consider the problem of learning a one-hidden-layer neural network: we assume the input x∈Rd is from Gaussian distribution and the label y=a⊤σ(Bx)+ξ, where a is a nonnegative vector in Rm with m≤d, B∈Rm×d is a full-rank weight matrix, and ξ is a n…
The paper analyzes convergence properties of NGA and PAMe for L1-norm PCA.
problem Finite-step convergence of L1-norm PCA algorithms. method Conditional subgradient and alternating maximization interpretations of NGA, and PAMe with extrapolation.
result Iterative points of modified NGA and PAMe remain constant after finitely many steps under certain conditions.
The paper addresses calibration in label ranking, a structured prediction task.
problem Calibration in label ranking is not well understood and often poorly calibrated.
method Formalized calibration for label ranking, developed a hierarchy of notions, and empirically evaluated models.
result Popular label ranking models are often poorly calibrated, with differences between sub-ranking and top-k metrics.
A necessary condition for a connection in a vector bundle to be locally metric is for its curvature matrix, which consists of 2 forms, to be skew symmetric with respect to some local frame. In this paper we give a simple algorithm that can be used to decide when a matrix of 2 forms is equivalent to a skew symmetric…
In this letter, we propose a new identification criterion that guarantees the recovery of the low-rank latent factors in the nonnegative matrix factorization (NMF) model, under mild conditions. Specifically, using the proposed criterion, it suffices to identify the latent factors if the rows of one factor are \emph{suf…
Determinantal point processes (DPPs) are an elegant model for encoding probabilities over subsets, such as shopping baskets, of a ground set, such as an item catalog. They are useful for a number of machine learning tasks, including product recommendation. DPPs are parametrized by a positive semi-definite kernel matrix…
SVD training reduces DNN rank and computation load without SVD per step.
problem High memory and computational load in deep neural networks.
method Explicitly achieves low-rank DNNs during training without SVD per step, using orthogonality regularization and sparsity-inducing regularizers.
result Significantly reduces DNN rank and computation load compared to existing methods.
Compressing DNNs is important for the real-world applications operating on resource-constrained devices. However, we typically observe drastic performance deterioration when changing model size after training is completed. Therefore, retraining is required to resume the performance of the compressed models suitable for…
Algorithm learns two-layer residual units using ReLU activations from samples.
problem Learning two-layer residual units from samples.
method Design layer-wise objectives as functionals, formulate ERM as QP, solve using LP, prove statistical consistency.
result Strong statistical consistency and robustness of the algorithm.
Optimal smooth subspaces approximate large data sets efficiently.
problem Approximating large data sets with invariant subspaces.
method Smooth functions under lattice translations or crystallographic groups, with optimal selection of Paley-Wiener space.
result Optimal lattice selection enhances approximation efficiency.
Sparse codes improve optimal control tasks with correlated inputs.
problem Optimal control tasks with correlated feature inputs.
method Used a sparse code to represent natural images in an optimal control task solved with neuro-dynamic programming.
result An over-complete sparse code increases memory capacity and learning speed beyond a complete code.