3 research outputs found
Efficient Second-Order Shape-Constrained Function Fitting
We give an algorithm to compute a one-dimensional shape-constrained function
that best fits given data in weighted- norm. We give a single
algorithm that works for a variety of commonly studied shape constraints
including monotonicity, Lipschitz-continuity and convexity, and more generally,
any shape constraint expressible by bounds on first- and/or second-order
differences. Our algorithm computes an approximation with additive error
in time, where
captures the range of input values. We also give a simple greedy algorithm that
runs in time for the special case of unweighted convex
regression. These are the first (near-)linear-time algorithms for
second-order-constrained function fitting. To achieve these results, we use a
novel geometric interpretation of the underlying dynamic programming problem.
We further show that a generalization of the corresponding problems to directed
acyclic graphs (DAGs) is as difficult as linear programming.Comment: accepted for WADS 2019; (v2 fixes various typos
Efficient Second-Order Shape-Constrained Function Fitting
We give an algorithm to compute a one-dimensional shape-constrained function that best fits given data in weighted- norm. We give a single algorithm that works for a variety of commonly studied shape constraints including monotonicity, Lipschitz-continuity and convexity, and more generally, any shape constraint expressible by bounds on first- and/or second-order differences. Our algorithm computes an approximation with additive error in time, where captures the range of input values. We also give a simple greedy algorithm that runs in time for the special case of unweighted convex regression. These are the first (near-)linear-time algorithms for second-order-constrained function fitting. To achieve these results, we use a novel geometric interpretation of the underlying dynamic programming problem. We further show that a generalization of the corresponding problems to directed acyclic graphs (DAGs) is as difficult as linear programming
A Nearly-Linear Time Algorithm for Linear Programs with Small Treewidth: A Multiscale Representation of Robust Central Path
Arising from structural graph theory, treewidth has become a focus of study
in fixed-parameter tractable algorithms in various communities including
combinatorics, integer-linear programming, and numerical analysis. Many NP-hard
problems are known to be solvable in time, where is the treewidth of the input
graph. Analogously, many problems in P should be solvable in time; however, due to the lack of appropriate tools,
only a few such results are currently known. [Fom+18] conjectured this to hold
as broadly as all linear programs; in our paper, we show this is true:
Given a linear program of the form , and a width- tree decomposition of a graph related to , we
show how to solve it in time where is the number of variables and is
the relative accuracy. Combined with recent techniques in vertex-capacitated
flow [BGS21], this leads to an algorithm with run-time. Besides being the first of its
kind, our algorithm has run-time nearly matching the fastest run-time for
solving the sub-problem (under the assumption that no fast matrix
multiplication is used).
We obtain these results by combining recent techniques in interior-point
methods (IPMs), sketching, and a novel representation of the solution under a
multiscale basis similar to the wavelet basis