Top-N-Rank improves top N item recommendations in scalable recommender systems.
problem Improving top N item recommendations in scalable recommender systems.
method Proposes a novel list-wise Learning-to-Rank model optimizing a variant of DCG objective function, incorporating weights for implicit feedback.
result Significant improvement in ranking quality for top N recommendations.
Improved understanding of low-rank solutions in SDPs via smoothed analysis.
problem Finding low-rank solutions to semidefinite programs efficiently.
method Penalty function formulation and smoothed analysis to avoid worst-case matrices.
result All approximate local optima are global optima for rank-constrained SDPs under certain conditions.
In binary classification and regression problems, it is well understood that Lipschitz continuity and smoothness of the loss function play key roles in governing generalization error bounds for empirical risk minimization algorithms. In this paper, we show how these two properties affect generalization error bounds in …
New algorithms improve transduction accuracy with low-rank data matrices.
problem Improving transduction accuracy with low-rank data matrices.
method Proposes two new algorithms using Smoothed Rank Function for transduction with Matrix Completion.
result Proposed methods outperform state-of-the-art methods in accuracy, especially in low observation rates.
StochasticRank optimizes ranking metrics efficiently and guarantees global convergence.
problem Optimizing discrete ranking metrics due to their ill-posed nature.
method Stochastic smoothing, gradient estimate, debiasing, and Stochastic Gradient Langevin Boosting.
result Global convergence and superior performance on ranking datasets.
A new federated learning algorithm improves on existing methods by exploiting data smoothness.
problem Federated learning optimization with smooth loss functions.
method Federated Low Rank Gradient Descent (FedLRGD) algorithm.
result FedLRGD outperforms Federated Averaging (FedAve) in federated oracle complexity under certain conditions.
New batch learning framework improves scalability and accuracy of personalized ranking.
problem Inaccurate rank estimation in large-scale personalized ranking algorithms.
method Uses batch-based rank estimators and smooth rank-sensitive loss functions.
result Consistent accuracy improvements and time efficiency advantages over state-of-the-art methods.
Matrix completion works well for smooth non-linear structures, even without low-rank assumptions.
problem Matrix completion for smooth non-linear structures.
method Nuclear-norm penalization for matrices lying in a low-dimensional non-linear manifold.
result Nuclear-norm penalization is minimax rate optimal for recovering smooth non-linear matrices with missing data.
New method extends low-rank MDPs to continuous action spaces.
problem Limited applicability of current low-rank MDP methods to continuous action spaces.
method Extending FLAMBE algorithm to continuous action spaces with Hölder smoothness conditions.
result Similar PAC bound achieved for continuous actions with polynomial dependence on smoothness order.
Ranking is a key aspect of many applications, such as information retrieval, question answering, ad placement and recommender systems. Learning to rank has the goal of estimating a ranking model automatically from training data. In practical settings, the task often reduces to estimating a rank functional of an object …
It is the main goal of this article to address the bipartite ranking issue from the perspective of functional data analysis (FDA). Given a training set of independent realizations of a (possibly sampled) second-order random function with a (locally) smooth autocorrelation structure and to which a binary label is random…
A new method for quantized matrix completion using Huber loss.
problem Quantized Matrix Completion with robustness to quantization errors.
method Rank minimization with Huber loss regularization, Smooth Rank Approximation.
result Our method achieves better accuracy and efficiency than state-of-the-art methods.
Derives smooth homogeneous structures for low-rank tensors.
problem Understanding the geometry of low-rank tensors.
method Analyzes sets of fixed CP, multilinear, and TT rank tensors to derive smooth homogeneous manifolds.
result Derives Riemannian metrics with complete geodesics.
Proposes a new method for multivariate functional regression.
problem Multivariate functional regression with complex relationships.
method Nested reduced-rank regularization (NRRR) approach.
result Consistent and effective in fitting multivariate functional regression models.
This work presents a general framework for solving the low rank and/or sparse matrix minimization problems, which may involve multiple non-smooth terms. The Iteratively Reweighted Least Squares (IRLS) method is a fast solver, which smooths the objective function and minimizes it by alternately updating the variables an…
Develops PRPCA for smooth image recovery combining low-rank and smoothness.
problem Image matrix recovery under low-rank and smoothness assumptions.
method Projected Robust PCA (PRPCA) framework combining low-rank and smoothness.
result Explicit statistical guarantees for PRPCA, reducing matrix dimensionality.
In this paper are given explicit calculations of Laplace operator spectrum for smooth real/complex-valued functions on all connected compact simple rank three Lie groups with biinvariant Riemannian metric and established a connection of obtained formulas with the number theory and integer ternary and binary quadratic f…
Paper proposes fast, robust methods for low-rank matrix recovery.
problem Estimating low-rank matrices from incomplete or corrupted data.
method Scaled subgradient methods for nonsmooth, nonconvex formulations.
result Methods converge almost dimension-free and condition-number independent.
New method solves nonsmooth low-rank matrix optimization problems efficiently.
problem Nonsmooth and low-rank matrix optimization problems in statistics and machine learning.
method Low-rank Extragradient Method with warm-start initialization.
result The extragradient method converges to an optimal solution with rate O(1/t) and requires only two low-rank SVDs per iteration. Paper develops fast low-rank approximation for smoothing splines.
problem Computational infeasibility of fitting cubic smoothing splines to large datasets.
method Low-rank approximation using eigensystem truncation.
result The method provides accurate, fast estimates with error bounds.
Low-rank matrix is desired in many machine learning and computer vision problems. Most of the recent studies use the nuclear norm as a convex surrogate of the rank operator. However, all singular values are simply added together by the nuclear norm, and thus the rank may not be well approximated in practical problems. …
A new algorithm improves both computational efficiency and statistical optimality for robust low-rank matrix and tensor estimation.
problem Challenges in low-rank matrix estimation under heavy-tailed noise, both computationally and statistically.
method Riemannian sub-gradient (RsGrad) algorithm, which is computationally efficient and statistically optimal.
result RsGrad achieves linear convergence and statistical optimality for robust loss functions under Gaussian and heavy-tailed noise.
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.
Satellite operations have infinite rank in smooth concordance group.
problem Understanding satellite operations in the smooth concordance group.
method Reduction to winding number zero satellites and use of SO(3) gauge theory. result Provides a criterion for satellite operations to generate infinite rank subgroups.
We generalize stochastic smoothing for gradient estimation of non-differentiable functions.
problem Gradient estimation for non-differentiable functions.
method Developed a general framework for relaxation and gradient estimation of non-differentiable black-box functions using stochastic smoothing with reduced assumptions.
result Empirically validated the effectiveness of variance reduction strategies for various non-differentiable tasks.
Many popular statistical models, such as factor and random effects models, give arise a certain type of covariance structures that is a summation of low rank and sparse matrices. This paper introduces a penalized approximation framework to recover such model structures from large covariance matrix estimation. We propos…
The paper studies invariants of differential 2-forms on manifolds.
problem Understanding invariants of differential 2-forms on manifolds.
method Analyzes the algebra of invariant functions on the bundle of maximal rank 2-covectors.
result The algebra of invariant functions is isomorphic to a specific algebra of invariant functions on the cotangent bundle.
It is of increasing importance to develop learning methods for ranking. In contrast to many learning objectives, however, the ranking problem presents difficulties due to the fact that the space of permutations is not smooth. In this paper, we examine the class of rank-linear objective functions, which includes popular…
Estimates Markov chains from data with a non-convex rank-constrained approach.
problem Estimating low-rank Markov chains from empirical data.
method Rank-constrained likelihood maximization and DC programming.
result The proposed estimator achieves better empirical performance than other methods.
Explores local structure of morphisms and formal submanifolds in formal manifolds theory.
problem Understanding the local structure of morphisms and formal submanifolds in formal manifolds.
method Study of formal manifolds, including local structure of constant rank morphisms and formal submanifolds.
result Developed the local structure of constant rank morphisms and formal submanifolds.
Improved Frank-Wolfe method tackles nonsmooth functions.
problem Efficiently solving large nonsmooth problems with sparse structures.
method Optimizes for approximation quality over all affine approximations.
result Overcomes issues with existing nonsmooth methods in low-rank matrix estimation.
In this paper are given explicit calculations of Laplace operator spectrum for smooth real/complex-valued functions on all connected compact simple rank four Lie groups with biinvariant Riemannian metric, corresponding to root systems B4, C4, D4 and established a connection of obtained formulas with the number…
Formula conjectured for refined SU(3) Vafa-Witten invariants of surfaces.
problem Calculating refined SU(3) Vafa-Witten invariants for smooth surfaces.
method Proved modularity transformation and used Mochizuki's formula and Maulik-Thomas's definition.
result Conjectured formula satisfies refined S-duality and verified in examples.
GLSKF improves tensor completion by capturing both global and local variations.
problem Tensor completion with missing entries, especially in data with spatial or temporal side information.
method Integrates smoothness-constrained low-rank factorization with a locally correlated residual process.
result GLSKF achieves superior performance and scalability on real-world datasets.
UA-LQE improves value function learning by selectively erasing uncertain entries in Q-matrix.
problem Improving value function learning in complex reinforcement learning tasks.
method Uncertainty-aware low-rank Q-matrix estimation (UA-LQE) algorithm.
result UA-LQE selectively erases uncertain entries in Q-matrix to improve value function approximation.
New concept of attitude towards probability introduced in risk sharing problems.
problem Risk sharing problems and attitudes towards probability.
method Generalized definition of probability premium, local approximation, rank-dependent utility model, dual theory.
result Attitude towards probability can be first-order or second-order, depending on the model.
SFM generates smooth functional data without exposing real data.
problem Challenges in statistical analysis of functional data.
method Copula framework and smooth flow construction.
result SFM produces high-quality synthetic functional data.
Matrix rank minimizing subject to affine constraints arises in many application areas, ranging from signal processing to machine learning. Nuclear norm is a convex relaxation for this problem which can recover the rank exactly under some restricted and theoretically interesting conditions. However, for many real-world …
Recent developments in linear system identification have proposed the use of non-parameteric methods, relying on regularization strategies, to handle the so-called bias/variance trade-off. This paper introduces an impulse response estimator which relies on an ℓ2-type regularization including a rank-penalty derive…
New algorithm for active bipartite ranking with continuous distributions.
problem Active ranking of bipartite data with continuous conditional distributions.
method Developed a novel algorithm called smooth-rank to minimize the distance between estimated and optimal ROC curves.
result Smooth-rank algorithm is PAC-(ε,δ) and outperforms existing methods in empirical tests. Tree tensor networks balance model complexity and empirical risk for high-dimensional function approximation.
problem Selecting optimal tree structure and ranks for high-dimensional function approximation.
method Proposes a complexity-based model selection method for tree tensor networks in empirical risk minimization.
result Demonstrates near-minimax adaptive performance across various smoothness classes.
The symmetry-rank of a riemannian manifold is by definition the rank of its isometry group. We determine precisely which smooth closed manifolds admit a positively curved metric with maximal symmetry-rank.
We consider the problem of learning multi-ridge functions of the form f(x) = g(Ax) from point evaluations of f. We assume that the function f is defined on an l_2-ball in R^d, g is twice continuously differentiable almost everywhere, and A \in R^{k \times d} is a rank k matrix, where k << d. We propose a randomized, po…
Gradient descent biases towards stable rank networks for nearly-orthogonal data.
problem Understanding implicit bias in non-smooth neural networks trained by gradient descent.
method Analysis of two-layer ReLU and leaky ReLU networks trained by gradient descent on nearly-orthogonal data.
result Gradient descent biases towards networks with stable rank and uniform margin for nearly-orthogonal data.
In many applications that require matrix solutions of minimal rank, the underlying cost function is non-convex leading to an intractable, NP-hard optimization problem. Consequently, the convex nuclear norm is frequently used as a surrogate penalty term for matrix rank. The problem is that in many practical scenarios th…
In many applications that require matrix solutions of minimal rank, the underlying cost function is non-convex leading to an intractable, NP-hard optimization problem. Consequently, the convex nuclear norm is frequently used as a surrogate penalty term for matrix rank. The problem is that in many practical scenarios th…
Solves geometric Cauchy problem for submanifolds with constant rank.
problem Finding submanifolds with constant rank in a given distribution.
method Constructive approach to solve the geometric Cauchy problem.
result A solution exists and is unique in a neighborhood of the submanifold.
New examples show satellite operations can expand the concordance group in topological knot theory.
problem Understanding how satellite operations affect the concordance group in topological knot theory.
method Forming satellites of knots with a fixed pattern and analyzing the induced map on the concordance group.
result Similar examples of rank-expanding satellite operations exist in the topological locally flat concordance group.