Max-linear regression problem solved with convex programming.
problem Estimating parameters in max-linear regression models.
method Formulated and analyzed a scalable convex program called anchored regression (AR).
result AR provides high probability recovery of parameters with a sample complexity of k4p. New solver MPLP++ outperforms existing solvers for dense graph models.
problem Efficiently solving dense, discrete Graphical Models with pairwise potentials.
method Dual Block-Coordinate Ascent with MPLP++ modification.
result MPLP++ significantly outperforms existing solvers, including TRWS.
Finding the most probable assignment (MAP) in a general graphical model is known to be NP hard but good approximations have been attained with max-product belief propagation (BP) and its variants. In particular, it is known that using BP on a single-cycle graph or tree reweighted BP on an arbitrary graph will give the …
Unified approach for optimizing predictions in linear programming and inverse problems.
problem Optimizing predictions in linear programming and inverse problems.
method Maximum optimality margin approach.
result Unified approach that balances computational efficiency and theoretical properties.
A new algorithm solves semidefinite programs using Langevin diffusion.
problem Optimizing semidefinite programs with diagonal constraints.
method Langevin diffusion on a product manifold of spheres.
result Langevin algorithm achieves ε accuracy in Ω(ε^-5) iterations.
PMP improves sampling and learning in complex energy models.
problem Intractability of MAP computation in EBMs.
method Perturb-and-max-product (PMP) for parallel and scalable sampling and learning.
result PMP outperforms existing methods in various models, including Ising and RBMs.
Unified approach to path planning using probabilistic inference on factor graphs.
problem Path planning problems using probabilistic inference.
method Unified framework using probabilistic factor graphs and message composition rules.
result Unified approach includes various algorithms like Sum-product, Max-product, Dynamic programming, and mixed criteria.
We propose a faster and more accurate method for learning classification trees.
problem Learning optimal binary classification trees is challenging and slow.
method We introduce a stronger MIP formulation and Benders' decomposition method.
result Our method is 50 times faster and improves out-of-sample performance.
A new portfolio optimization model minimizes maximum drawdown, offering faster and more robust solutions.
problem Optimizing portfolios during financial distress, especially during crises.
method Linearization of Markowitz model based on maximum drawdown, with a Mixed-Integer Linear Programming variation.
result 200 times faster solving time with a more profitable and robust solution.
In this paper we propose an approach to preference elicitation that is suitable to large configuration spaces beyond the reach of existing state-of-the-art approaches. Our setwise max-margin method can be viewed as a generalization of max-margin learning to sets, and can produce a set of "diverse" items that can be use…
This paper addresses a novel data science problem, prescriptive price optimization, which derives the optimal price strategy to maximize future profit/revenue on the basis of massive predictive formulas produced by machine learning. The prescriptive price optimization first builds sales forecast formulas of multiple pr…
Let SL(n,Z) be the special linear group over integers and M=S1r×S2r,T1r×S2r , or T0r×S1r×S2r, products of spheres and tori. We prove that any group action of SL(n,Z) on Mr by diffeomorphims or piecewise linear homeomorphisms is trivial if r<n−1. This confirms a conjec…
Novel approximation hierarchy for sparse quadratic programs.
problem Sparse Quadratic Programs with Cardinality Constraints.
method Exploits rank-dominating eigenvectors for min-max optimization over binary variables.
result Efficient screening of nonzero elements with scalable optimization algorithms.
Observations depending on sums of random variables are common throughout many fields; however, no efficient solution is currently known for performing max-product inference on these sums of general discrete distributions (max-product inference can be used to obtain maximum a posteriori estimates). The limiting step to …
New method for finding function correspondences in binary programs.
problem Finding a correspondence between functions of two binary programs.
method Formulated as a network alignment problem and solved using belief propagation.
result QBinDiff prototype outperforms state-of-the-art diffing tools.
New algorithm clusters Gaussian mixtures with unknown covariance efficiently.
problem Clustering data from a mixture of Gaussians with unknown covariance.
method Developed an efficient spectral algorithm based on a Max-Cut integer program.
result Achieves optimal misclassification rate with quadratic sample size.
A recent trend observed in traditionally challenging fields such as computer vision and natural language processing has been the significant performance gains shown by deep learning (DL). In many different research fields, DL models have been evolving rapidly and become ubiquitous. Despite researchers' excitement, unfo…
Tropical SVM tackles phylogenomics by classifying multi-locus data.
problem Classifying multi-locus data sets for phylogenetic analysis.
method Proposes tropical support vector machines (SVMs) for phylogenomics, formulated as linear programming problems.
result Developed methods for hard and soft margin tropical SVMs, proving necessary and sufficient conditions for separation.
Study on stable torsion length in groups, showing it vanishes in crystallographic groups and providing algorithms for computation.
problem Understanding the stable torsion length in groups, especially in crystallographic and free products of groups.
method Developed linear programming and exact algorithms to compute stable torsion length in free products of groups and finite groups.
result Showed that stable torsion length vanishes in crystallographic groups and provided exact computations for nontrivial examples.
In 1960s, Almgren initiated a program to find minimal hypersurfaces in compact manifolds using min-max method. This program was largely advanced by Pitts and Schoen-Simon in 1980s when the manifold has no boundary. In this paper, we finish this program for general compact manifold with nonempty boundary. As a result, w…
New method uses DC functions for piecewise linear regression.
problem Regression with piecewise linear constraints.
method Estimates piecewise linear convex functions using a difference of convex functions.
result Method achieves close to minimax statistical risk and comparable performance to existing methods.
Estimates parameters in max-linear Bayesian networks with noise.
problem Causal inference in extreme-value settings with noise parameters.
method Max-plus algebra and logarithm transformation, normal distribution estimation, EM algorithm and quadratic optimization.
result An estimator of a parameter for each edge in a DAG is normally distributed.
This paper shows neural networks can solve complex graph problems efficiently.
problem Solving exact maximum flow computation and minimum spanning tree problems.
method Introduces Max-Affine Arithmetic Programs and shows equivalence to neural networks.
result Two combinatorial optimization problems can be solved with polynomial-size neural networks.
In this paper, we prove that the Morse index of a multiplicity one, smooth, min-max minimal hypersurface is generically equal to the dimension of the homology class detected by the families used in the construction. This confirms part of the program (\cite{marques-icm}, \cite{marques-neves-cycles}, \cite{marques-neves-…
The paper explores optimal algorithms for linear regression under covariate shift, proving the optimality of certain transformations and SGD variants.
problem Optimal algorithms for linear regression under covariate shift with ellipse-shaped constraints.
method Establishes a tight lower generalization bound via Bayesian Cramer-Rao inequality, proves the optimality of certain transformations, and analyzes SGD variants.
result Optimal estimators and SGD variants achieve optimality under specific conditions.
Integral geometry explores curvature conjectures on manifolds.
problem Proving curvature conjectures for manifolds of positive and negative curvature.
method Formulating curvature integral geometrically and using linear programming.
result Existence of a positive curvature metric on product manifolds.
The study counts minimal surfaces in 3-manifolds with positive Ricci curvature.
problem Counting minimal surfaces in 3-manifolds with positive Ricci curvature.
method An enumerative min-max theorem linking surface counts to topological properties.
result Every 3-sphere of positive Ricci curvature contains at least 4 embedded minimal surfaces of genus 2.
Generalizes leverage score sampling for neural networks, accelerating kernel methods and deep learning.
problem Accelerating kernel methods and deep learning training.
method Generalizes leverage score sampling to neural networks and proves equivalence to neural tangent kernel ridge regression.
result Equivalence between regularized neural network and neural tangent kernel ridge regression under leverage score sampling initialization.
SPPL simplifies probabilistic programming for exact inference.
problem Efficient exact inference in probabilistic models.
method SPPL translates probabilistic programs into sum-product expressions, leveraging new techniques for scalability.
result SPPL achieves up to 3500x speedups in exact inference.
We reformulate LIPs as min-max problems for easier solution.
problem Recovering signals from few linear measurements.
method Proposed a min-max reformulation of LIPs.
result Saddle points characterize solutions to LIPs.
New model optimizes oil product distribution via pipelines.
problem Optimizing oil product distribution via pipelines.
method Discrete-time mixed integer linear programming model.
result Significant reductions in pipeline operational cost.
The study finds a way to create minimal surfaces with specific shapes in 3-manifolds.
problem Creating minimal surfaces with prescribed genus in 3-manifolds with positive Ricci curvature.
method Develops a min-max theory and shows deformability of surfaces in a generic metric.
result Establishes a theorem for producing minimal surfaces with prescribed genus.
The paper proposes a machine learning technique to optimize prices in fashion e-commerce.
problem Optimizing prices for millions of products in fashion e-commerce to maximize revenue and profit.
method Demand prediction, price elasticity, multiple price demand pairs, linear programming optimization.
result The model improved revenue by 1% and gross margin by 0.81% in AB tests.
Paper presents ABGD for efficient piecewise linear regression in high dimensions.
problem Efficiently solving piecewise linear regression in high-dimensional spaces.
method Parametrizes piecewise linear functions as difference of max-affine functions, using ABGD algorithm.
result ABGD converges linearly to an ε-accurate estimate with optimal sample complexity.
We prove the formula TC(G∗H)=max{TC(G),TC(H),cd(G×H)} for the topological complexity of the free product of discrete groups with cohomological dimension >2.
Max-product Belief Propagation (BP) is a popular message-passing algorithm for computing a Maximum-A-Posteriori (MAP) assignment over a distribution represented by a Graphical Model (GM). It has been shown that BP can solve a number of combinatorial optimization problems including minimum weight matching, shortest path…
A new reinforcement learning method improves Max-Cut solutions without needing training data.
problem Max-Cut problem is NP-hard, and existing methods struggle with generalizability and scalability.
method Training-data-free reinforcement learning approach to hyperplane rounding for Max-Cut optimization.
result Our method consistently achieves better Max-Cut solutions across various graph types.
Max-affine regression method converges linearly using GD and SGD.
problem Regression of max-affine models in signal processing and statistics.
method Gradient descent and mini-batch stochastic gradient descent analysis.
result GD and SGD converge linearly to a neighborhood of the ground truth under sub-Gaussian assumptions.
The closed string field theory minimal-area problem asks for the conformal metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length at least 2π. This is an extremal length problem in conformal geometry as well as a problem in systolic geometry. We consider the ana…
The min-max kernel is a generalization of the popular resemblance kernel (which is designed for binary data). In this paper, we demonstrate, through an extensive classification study using kernel machines, that the min-max kernel often provides an effective measure of similarity for nonnegative data. As the min-max ker…
Tropical Geometry and Mathematical Morphology share the same max-plus and min-plus semiring arithmetic and matrix algebra. In this chapter we summarize some of their main ideas and common (geometric and algebraic) structure, generalize and extend both of them using weighted lattices and a max-⋆ algebra with an ar…
We propose a Bayesian expectation-maximization (EM) algorithm for reconstructing Markov-tree sparse signals via belief propagation. The measurements follow an underdetermined linear model where the regression-coefficient vector is the sum of an unknown approximately sparse signal and a zero-mean white Gaussian noise wi…
Efficiently finds sparse solutions to max-plus equations for convex regression.
problem Finding sparse solutions to max-plus equations for convex multivariate regression.
method Polynomial-time algorithm for sparse approximate solutions.
result Optimal piecewise-linear fitting with minimum number of regions.
MAP inference for general energy functions remains a challenging problem. While most efforts are channeled towards improving the linear programming (LP) based relaxation, this work is motivated by the quadratic programming (QP) relaxation. We propose a novel MAP relaxation that penalizes the Kullback-Leibler divergence…
The paper finds optimal ways to combine ETFs to minimize costs for investors.
problem Finding the best combination of ETFs to match a target gearing ratio at the lowest expense.
method Linear programming and convex geometry to prove the two-fund theorem for ETFs.
result The cheapest way to achieve a target gearing ratio is by combining the two nearest undominated ETF products.
Framework learns linear programs from optimal decisions.
problem Learning linear programs from optimal decisions is challenging.
method Gradient-based framework for learning linear programs from optimal decisions.
result Successfully learns linear programs and multi-commodity flow instances.
Convex message passing algorithms converge to a fixed point.
problem Understanding convergence properties of convex message passing methods.
method Proving convergence of coordinate descent applied to piecewise-affine convex objectives, and showing this applies to various message passing methods.
result The iterates converge to a fixed point of the method, and the algorithm terminates in a known number of iterations.
New findings on the max margin problem in neural networks.
problem Understanding the max margin problem in neural networks.
method Analyzing gradient flow and max margin problem in linear and ReLU networks.
result The KKT point is not always an optimum of the max margin problem.