Study shows non-symmetric convex sets have full boundary limits.
problem Understanding boundaries of non-symmetric convex sets.
method Proved using proximal limit set analysis.
result Proximal limit set equals full projective boundary for non-symmetric irreducible divisible convex sets.
Large sectors of the recent optimization literature focused in the last decade on the development of optimal stochastic first order schemes for constrained convex models under progressively relaxed assumptions. Stochastic proximal point is an iterative scheme born from the adaptation of proximal point algorithm to nois…
Two strategies for embedding new data points from proximity data are explored.
problem Embedding new data points using proximity data.
method Two competing strategies: projection and restricted reconstruction.
result Projection and restricted reconstruction can be derived from kernel methods.
New method decomposes local projections to reveal historical drivers of estimates.
problem Uncertainty in interpreting local projections due to black-box nature.
method Decomposes LP estimates into contributions of historical events, interpreting weights as shocks and proximity scores.
result Dominant historical events drive impulse response estimates, revealing underlying mechanisms.
A new method for sparse regression models using graph structure.
problem Sparse regression models for high-dimensional data.
method Decomposes coefficient vector into latent variables, performs regularization on latent variables, uses proximal projection.
result Stable performance compared to other models, especially for high-dimensional data.
We develop a projected Nesterov's proximal-gradient (PNPG) approach for sparse signal reconstruction that combines adaptive step size with Nesterov's momentum acceleration. The objective function that we wish to minimize is the sum of a convex differentiable data-fidelity (negative log-likelihood (NLL)) term and a conv…
Low-rank inducing unitarily invariant norms have been introduced to convexify problems with low-rank/sparsity constraint. They are the convex envelope of a unitary invariant norm and the indicator function of an upper bounding rank constraint. The most well-known member of this family is the so-called nuclear norm. To …
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.
Paper examines convergence rate of PGD for BP objective in inverse problems.
problem Optimizing ill-posed linear inverse problems using BP vs LS.
method Analysis of PGD convergence rate for BP objective, comparison with proximal gradient method.
result PGD converges faster for BP objective due to inherent properties.
Network embedding, which learns low-dimensional vector representation for nodes in the network, has attracted considerable research attention recently. However, the existing methods are incapable of handling billion-scale networks, because they are computationally expensive and, at the same time, difficult to be accele…
Wasserstein distance plays increasingly important roles in machine learning, stochastic programming and image processing. Major efforts have been under way to address its high computational complexity, some leading to approximate or regularized variations such as Sinkhorn distance. However, as we will demonstrate, regu…
New algorithms accelerate model-based optimization for stochastic problems.
problem Optimizing model-based stochastic optimization problems efficiently.
method Proposed new model-based algorithms with acceleration and minibatch techniques.
result Non-asymptotic convergence guarantees with linear speedup in minibatch size.
Low-rank modeling has a lot of important applications in machine learning, computer vision and social network analysis. While the matrix rank is often approximated by the convex nuclear norm, the use of nonconvex low-rank regularizers has demonstrated better recovery performance. However, the resultant optimization pro…
Fix a finite set of points in Euclidean n-space $\euc^n$, thought of as a point-cloud sampling of a certain domain $D\subset\euc^n$. The Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional approximation to the homotopy type of D. …
A new method solves l1-regularized optimization problems efficiently and sparsely.
problem l1-regularized optimization problems in machine learning.
method Orthant Based Proximal Stochastic Gradient Method (OBProx-SG)
result Promotes sparsity of solutions substantially and converges to global optimal solutions.
We develop a novel theoretical framework for understating OT schemes respecting a class structure. For this purpose, we propose a convex OT program with a sum-of-norms regularization term, which provably recovers the underlying class structure under geometric assumptions. Furthermore, we derive an accelerated proximal …
PPGD solves nonconvex nonsmooth optimization problems without KL property.
problem Nonconvex and nonsmooth optimization problems in statistics and machine learning.
method Projective Proximal Gradient Descent (PPGD) for solving a class of nonconvex and nonsmooth problems.
result PPGD achieves a fast convergence rate of O(1/k^2) for k ≥ k_0.
New sampling methods for constrained and composite distributions.
problem Sampling from log-concave distributions with constraints and composite structures.
method Proximal sampler applied to lifted convex sets with separation and subgradient oracles.
result Practical and unbiased samplers for constrained and composite distributions.
We consider stochastic strongly convex optimization with a complex inequality constraint. This complex inequality constraint may lead to computationally expensive projections in algorithmic iterations of the stochastic gradient descent~(SGD) methods. To reduce the computation costs pertaining to the projections, we pro…
Characterizes minimizing curves in Riemannian manifolds.
problem Finding optimal paths in curved spaces.
method Characterization of prox-regular sets and tangent cones.
result Necessary condition for minimizing curves in prox-regular sets.
In machine learning research, the proximal gradient methods are popular for solving various optimization problems with non-smooth regularization. Inexact proximal gradient methods are extremely important when exactly solving the proximal operator is time-consuming, or the proximal operator does not have an analytic sol…
Improves time series classification with forest proximities.
problem Time series classification accuracy and efficiency.
method PF-GAP, an extension of RF-GAP proximities to proximity forests, combined with Multi-Dimensional Scaling and Local Outlier Factors.
result Forest proximities show stronger connection between misclassified points and outliers.
Improved sampling guarantees for weakly log-concave distributions.
problem Sampling from distributions that are not strongly log-concave.
method Proximal sampler with convergence guarantees under weaker assumptions.
result New state-of-the-art sampling guarantees for various target distributions.
New algorithms solve complex minimax problems without needing derivatives.
problem Solving nonconvex-concave minimax problems efficiently.
method Zeroth-order alternating and proximal gradient algorithms.
result Iteration complexity and function value estimation bounds established.
Paper analyzes complexity of PSGLA for sampling log-concave distributions.
problem Sampling from log-concave distributions with composite potentials.
method Uses primal-dual interpretation and duality gap to analyze PSGLA complexity.
result Complexity of PSGLA is O(1/ε2) for strongly convex potentials. CFR-Pro enhances treatment effect estimation by incorporating local proximity.
problem Treatment selection bias in HTE estimation from observational data.
method Proximity-enhanced CounterFactual Regression (CFR-Pro) with pair-wise proximity regularizer and subspace projector.
result Significantly outperforms competitors in HTE estimation accuracy.
Improved random forest proximities capture data geometry.
problem Inaccurate random forest proximities do not reflect learned data geometry.
method Introduce RF-GAP: Geometry- and Accuracy-Preserving proximities.
result RF-GAP improves geometric representation in tasks like data imputation.
Introduces PPMM algorithm for nonconvex robust regression problems.
problem Nonconvex tuning-free robust regression problems.
method PPMM algorithm with inner subproblems solved by SSN-PPA.
result Converges to d-stationary point with KL property.
Paper analyzes convergence of proximal algorithm in metric spaces without geodesic convexity.
problem Analyzing convergence of proximal algorithm in general metric spaces.
method Analysis of the Wasserstein proximal algorithm without geodesic convexity assumption.
result Establishes unbiased and linear convergence rate for proximal algorithm under natural Wasserstein inequality.
Extends RF proximities to all supervised distance-based machine learning contexts.
problem Limited utility of RF proximities in various machine learning tasks.
method Introduces generalized Proximity Forest (PF) model and variant for regression.
result Demonstrates unique advantages over RF and k-nearest neighbors models.
Improved bounds for proximal gradient algorithms with computational errors.
problem Analyzing convergence of proximal gradient algorithms with inaccuracies.
method Deriving new tighter deterministic and probabilistic bounds for convex composite problems.
result Probabilistic bounds are more robust and accurate for algorithm verification and performance guarantees.
Paper extends theorem on covering spaces and Jordan curves.
problem Covering and extending theorems for Alexandrov spaces.
method Introduces proximal homotopic cycles to extend the Mitsuishi-Yamaguchi theorem.
result Extensions of the Mitsuishi-Yamaguchi Good Covering Theorem and Jordan curve theorem.
Proximal algorithms applied to current deformation into cycles.
problem Deformation of de Rham currents into cycles.
method Proximal algorithms, total variation denoising for differential forms.
result Calibrated cycles constructed in calibrated manifolds.
Innovative method solves nonconvex optimization on manifolds.
problem Nonconvex optimization problems on Riemannian manifolds.
method Intrinsic Riemannian proximal gradient method.
result Converges for nonconvex or nonembedded problems.
New PnP algorithm converges with relaxed proximal gradient descent.
problem Convergence issues in PnP methods with deep denoisers.
method Relaxed proximal gradient descent for PnP with weakly convex regularization.
result Proposed PnP-αPGD converges for a wider range of regularization parameters. EPINE enhances network embedding by improving adjacency matrix-based high-order proximity.
problem Inaccurate and poorly designed calculation of high-order proximity in network embedding.
method EPINE redefines high-order proximity intuitively and proposes a scalable algorithm for accurate calculation.
result EPINE outperforms existing methods in network reconstruction, link prediction, and node classification.
Stochastic version of proximal distance algorithm analyzed and validated.
problem Optimization of constrained estimation problems.
method Stochastic proximal distance algorithm, with convergence guarantees and finite error bounds.
result Convergence guarantees and finite error bounds for the first time.
This paper extends the Good Covering Theorem and Jordan Curve Theorem for proximal Alexandrov spaces.
problem Extending the Good Covering Theorem and Jordan Curve Theorem to proximal Alexandrov spaces.
method Introducing path cycles and using them to extend the Good Covering Theorem and Jordan Curve Theorem.
result Extensions of the Mitsuishi-Yamaguchi Good Covering Theorem and Jordan Curve Theorem for proximal Alexandrov spaces.
This paper studies fixed sets in ribbon complexes using descriptive proximity spaces.
problem Understanding fixed sets in ribbon complexes within descriptive proximity spaces.
method Introduces descriptive fixed sets and their properties in ribbon complexes, using descriptive proximally continuous maps.
result Establishes that proximal descriptive conjugacy preserves fixed sets in ribbon complexes.
Introduces a new divergence measure for optimal transport.
problem Optimal transport distances and information divergences.
method Infimal convolution formulation of proximal optimal transport divergence.
result Establishes connections to dynamic formulations and partial differential equations.
We propose a new proximal, path-following framework for a class of constrained convex problems. We consider settings where the nonlinear---and possibly non-smooth---objective part is endowed with a proximity operator, and the constraint set is equipped with a self-concordant barrier. Our approach relies on the followin…
Unified framework for training neural networks with non-smooth, non-convex regularizers.
problem Training neural networks with non-smooth, non-convex regularizers.
method ProxGen framework for stochastic proximal gradient descent.
result ProxGen framework achieves the same convergence rate as standard methods and outperforms subgradient-based approaches.
A new method reformulates Optimal Transport Conditional Flow Matching using proximal operators.
problem Optimal Transport Conditional Flow Matching (OT-CFM) for generating models.
method Reformulate OT-CFM using proximal operators and extended Brenier potential.
result OT-CFM dynamics are terminally normally hyperbolic for manifold-supported targets.
SMP model preserves proximity and permutation in graph neural networks.
problem Challenges in graph mining, such as community and leader finding.
method Stochastic Message Passing (SMP) model that maintains proximity and permutation-equivariance.
result SMP model effectively preserves node proximities and permutation-equivariance.
Many machine learning techniques sacrifice convenient computational structures to gain estimation robustness and modeling flexibility. However, by exploring the modeling structures, we find these "sacrifices" do not always require more computational efforts. To shed light on such a "free-lunch" phenomenon, we study the…
Paper relates asymptotic dimension to cofinal dimension using coarse proximities.
problem Relating asymptotic dimension to cofinal dimension in metric spaces.
method Introducing coarse proximities and inverse limit constructions.
result Asymptotic dimension is bounded by coarse cofinal dimension and cofinal dimension of Higson corona.
Proximal Diffusion Models improve generative model efficiency.
problem Improving generative model efficiency and accuracy.
method Developed Proximal Diffusion Models using proximal maps instead of scores.
result Proximal Diffusion Models achieve faster convergence and higher accuracy.
Deep neural networks improve proximal inference for causal effects.
problem Estimating causal effects in the presence of unmeasured confounders.
method Flexible deep neural network to estimate the bridge function.
result Achieves state-of-the-art performance on benchmarks.