Paper proposes a new method for SP with covariates using PADR and ERM.
problem Stochastic programming with covariate information.
method Empirical risk minimization (ERM) with nonconvex piecewise affine decision rules (PADR).
result The method provides theoretical consistency and computational tractability for nonconvex SP problems.
Neural networks can represent complex piecewise functions efficiently.
problem Representing continuous piecewise affine functions with neural networks.
method Two hidden layers with ReLU activation, O(p) neurons for p pieces. result CPA functions can be represented by a neural network with linear size.
We use partial actions, as formalized by Exel, to construct various commensurating actions. We use this in the context of groups piecewise preserving a geometric structure, and we interpret the transfixing property of these commensurating actions as the existence of a model for which the group acts preserving the geome…
BN refines local partition geometry in piecewise-affine networks during training.
problem Understanding the effect of BN on the function realized during training in piecewise-affine networks.
method Analyzing the geometry of switching hyperplanes and affine-region partition conditioned on a mini-batch.
result BN increases expected local partition refinement in ReLU and piecewise-affine networks.
In exchange for large quantities of data and processing power, deep neural networks have yielded models that provide state of the art predication capabilities in many fields. However, a lack of strong guarantees on their behaviour have raised concerns over their use in safety-critical applications. A first step to unde…
Batch normalization improves deep networks by aligning their decision boundaries with data.
problem Improving the performance and generalization of deep networks.
method Theoretical analysis of batch normalization as a function approximation technique for continuous piecewise affine splines.
result Batch normalization adapts the geometry of a deep network's partition to match the data, improving learning and generalization.
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.
Method identifies latent variables from high-dimensional data with piecewise affine mixing.
problem Identifying latent variables from high-dimensional observations with dependencies and piecewise affine transformations.
method Proposes a two-stage method with sparsity and Gaussianity regularization.
result Effectively recovers ground-truth latent variables from synthetic and image data.
Proposes a new decision rule for continuous treatments.
problem Developing personalized treatment recommendations for continuous treatments.
method Jump interval-learning method to estimate conditional mean of outcomes.
result Optimal interval-valued decision rule (I2DR) for continuous treatments.
This work uses tropical geometry to understand neural network decision boundaries.
problem Characterizing neural network decision boundaries with piecewise linear activations.
method Tropical geometry applied to a simple neural network model.
result Decision boundaries are a subset of a tropical hypersurface related to a polytope formed by zonotopes.
Paper develops algorithms for PWA systems with polynomial regret.
problem Learning in piecewise affine systems due to discontinuities.
method Smoothed online learning framework applied to PWA systems.
result First algorithms with polynomial regret in PWA systems.
We study the coarse geometry of the moduli space of dilation tori with two singularities and the dynamical properties of the action of the Teichmuller flow on this moduli space. This leads to a proof that the vertical foliation of a dilation torus is almost always Morse-Smale. As a corollary, we get that the generic pi…
We learn sensor trees from training data to minimize sensor acquisition costs during test time. Our system adaptively selects sensors at each stage if necessary to make a confident classification. We pose the problem as empirical risk minimization over the choice of trees and node decision rules. We decompose the probl…
Optimal Volt/VAR control rules designed using deep learning.
problem Designing optimal Volt/VAR control rules for DERs to regulate voltage fluctuations.
method Formulated as a deep learning problem, where a DNN emulates Volt/VAR dynamics and optimizes rule parameters.
result DNN-based optimization outperforms MINLP in efficiency and accuracy.
We study the geometry of deep (neural) networks (DNs) with piecewise affine and convex nonlinearities. The layers of such DNs have been shown to be {\em max-affine spline operators} (MASOs) that partition their input space and apply a region-dependent affine mapping to their input to produce their output. We demonstrat…
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.
Study growth patterns in random networks using i.i.d. perturbations.
problem Understanding the growth of affine regions in random piecewise-linear networks.
method Analyzes a random compositional model with i.i.d. perturbations of the tent map, proving submultiplicative pressure and using finite-state defect process for upper-tail lower bounds.
result Proves the existence of a submultiplicative pressure for \(N_n\) and gives exponential upper bounds for \(n^{-1}\log N_n\).
For high-dimensional classification, it is well known that naively performing the Fisher discriminant rule leads to poor results due to diverging spectra and noise accumulation. Therefore, researchers proposed independence rules to circumvent the diverse spectra, and sparse independence rules to mitigate the issue of n…
Non-affine aggregation rules cannot preserve monotonicity in convex learning.
problem Designing non-affine aggregation rules that maintain monotonicity in convex learning.
method Proving that monotonicity of aggregated gradients is preserved only if the aggregation rule is positively affine.
result Non-affine aggregation prevents steady convergence and substantially degrades algorithmic stability.
PAR provides a flexible framework for quantization in optimization problems.
problem Challenges in optimization problems over discrete or quantized variables.
method Piecewise-affine regularization (PAR) for modeling and computational optimization.
result PAR-regularized loss functions exhibit high quantization at critical points in the overparameterized regime.
This paper considers affine analogues of the isoperimetric inequality in the sense of piecewise linear topology. Given a closed polygon P embedded in R^d having n edges, we give upper and lower bounds for the minimal number of triangles needed to forma triangulated embedded orientable surface in R^d having P as its geo…
New insights into how neural networks classify data.
problem Understanding the topological structure of decision regions in ReLU networks.
method Defining generic and transversal ReLU networks, and using linear complexes to identify obstructions.
result Generic, transversal ReLU networks have at most one bounded connected component in their decision regions.
Binary classification is a common statistical learning problem in which a model is estimated on a set of covariates for some outcome indicating the membership of one of two classes. In the literature, there exists a distinction between hard and soft classification. In soft classification, the conditional class probabil…
The study classifies singularities in discrete improper affine spheres.
problem Classifying singularities in discrete improper affine spheres.
method Analysis of discrete improper affine spheres based on asymptotic nets, distinguishing singular edges and vertices.
result First step in classifying singularities of discrete nets.
Exact LAD line fitting via PALB with linear scaling and speed.
problem Robust line fitting for data with outliers.
method Piecewise Affine Lower-Bounding (PALB) method using supporting lines and subdivision scheme.
result Empirical log-linear scaling and significantly faster than LP and IRLS methods.
Defines CAMC discrete nets and their properties.
problem Understanding CAMC discrete nets and their properties.
method Defining CAMC discrete nets and proving properties.
result Properties of CAMC discrete nets are equivalent to properties of compatible interpolating quadrics.
Paper extends transfer learning for decision rules, improving treatment rule estimation.
problem Estimating optimal individualized treatment rules under changing conditions.
method Bayes decision rules and low-dimensional empirical risk minimization.
result Consistent estimators and risk bounds established under mild conditions.
The paper designs neural networks with assurance for controlling nonlinear systems.
problem Designing neural networks with assurance for nonlinear system control.
method Bounding the number of affine functions needed for a CPWA function, connecting it to a TLL NN architecture.
result The TLL NN architecture is parameterized by the number of affine functions in the CPWA function it realizes.
PARC uses piecewise linear predictors for regression and classification.
problem Multivariate regression and classification problems.
method Alternates between ridge and softmax regression, and cluster assignment based on accuracy and separability.
result Converges to a local minimum in a finite number of steps.
No policy can simultaneously be fully autonomous, optimally calibrated, and helpful, proving a trilemma.
problem Proving impossibility of a policy achieving maximum helpfulness, optimal calibration, and full autonomy.
method Geometric proof showing that adding any non-affine autonomy incentive to a strictly proper scoring rule destroys strict properness.
result The Behavioral Credibility Trilemma: no policy can achieve all three goals simultaneously.
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.
Nonlinearity is crucial to the performance of a deep (neural) network (DN). To date there has been little progress understanding the menagerie of available nonlinearities, but recently progress has been made on understanding the rôle played by piecewise affine and convex nonlinearities like the ReLU and absolute value …
We present a detailed analysis of the class of regression decision tree algorithms which employ a regulized piecewise-linear node-splitting criterion and have regularized linear models at the leaves. From a theoretic standpoint, based on Rademacher complexity framework, we present new high-probability upper bounds for …
We consider the quasiconformal dilatation of projective transformations of the real projective plane. For non-affine transformations, the contour lines of dilatation form a hyperbolic pencil of circles, and these are the only circles that are mapped to circles. We apply this result to analyze the dilatation of the circ…
The paper provides results regarding the computational complexity of hybrid system identification. More precisely, we focus on the estimation of piecewise affine (PWA) maps from input-output data and analyze the complexity of computing a global minimizer of the error. Previous work showed that a global solution could b…
New EM algorithm improves deep generative network training.
problem Training deep generative networks with complex posterior and likelihood distributions.
method Derive analytical posterior and marginal distributions using CPA property, derive analytical EM algorithm.
result EM training yields higher likelihood than Variational Autoencoders (VAEs).
New decision-theoretic characterization separates belief and decision posteriors.
problem Understanding the conditions under which loss-based updating coincides with Bayesian updating.
method Decision-theoretic approach to distinguish belief and decision posteriors.
result Generalized Bayes coincides with ordinary Bayesian updating only if the loss is proportional to negative log-likelihood.
Proposed by Donoho (1997), Dyadic CART is a nonparametric regression method which computes a globally optimal dyadic decision tree and fits piecewise constant functions in two dimensions. In this article we define and study Dyadic CART and a closely related estimator, namely Optimal Regression Tree (ORT), in the contex…
Optimal timing for converting savings into annuities considering mortality risk.
problem Determining the best time to annuitize retirement savings under stochastic mortality.
method Formulated as a three-dimensional optimal stopping problem, reduced to nested one-dimensional problems, solved using PDMP structure.
result Rich structure for the optimal annuitization rule, covering various parameter specifications.
A new method learns interpretable decision rules using submodular optimization.
problem Learning interpretable decision rules from data.
method Submodular optimization approach for selecting rules from a large set.
result The method effectively learns interpretable rule sets from real datasets.
From doctors diagnosing patients to judges setting bail, experts often base their decisions on experience and intuition rather than on statistical models. While understandable, relying on intuition over models has often been found to result in inferior outcomes. Here we present a new method, select-regress-and-round, f…
New algorithms optimize decision rules in strategic scenarios, minimizing prediction risk and incentivizing better outcomes.
problem Strategic agents manipulate features to improve outcomes, complicating decision-making models.
method Efficient algorithms for learning decision rules that minimize prediction risk, incentivize better outcomes, and estimate true model coefficients.
result Optimal decision rules can be learned through testing and observing agent responses, circumventing hardness results.
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.
Two algorithms for interpreting and boosting tree-based models using rule covering.
problem Interpreting and boosting tree-based ensemble methods.
method Mathematical programming models constructed from decision tree rules.
result Selects a few rules that closely match the accuracy of the model.
Basic aspects of the equiaffine geometry of level sets are developed systematically. As an application there are constructed families of 2n-dimensional nondegenerate hypersurfaces ruled by n-planes, having equiaffine mean curvature zero, and solving the affine normal flow. Each carries a symplectic structure with r…
Normalizing flows attempt to model an arbitrary probability distribution through a set of invertible mappings. These transformations are required to achieve a tractable Jacobian determinant that can be used in high-dimensional scenarios. The first normalizing flow designs used coupling layer mappings built upon affine …
LI-ITR combines flexible ML with interpretable approximations for personalized treatment rules.
problem Combining flexibility and interpretability in personalized treatment rules.
method Uses variational autoencoders and a mixture of interpretable experts.
result Accurately recovers true local coefficients and optimal treatment strategies.
We give a geometric description of the fusion rules of the affine Lie algebra su(2)_k at a positive integer level k in terms of the k-th power of the basic gerbe over the Lie group SU(2). The gerbe can be trivialised over conjugacy classes corresponding to dominant weights of su(2)_k via a 1-isomorphism. The fusion-rul…