arXiv subjects

Amitabh Basu

Publications and source records attributed to Amitabh Basu.

At least 55 records · Page 3Linked to original sources

arXiv2018

An extreme function which is nonnegative and discontinuous everywhere

We consider Gomory and Johnson's infinite group model with a single row. Valid inequalities for this model are expressed by valid functions and it has been recently shown that any valid function is dominated by some nonnegative valid function, modulo the affine hull of the model. Within the set of nonnegative valid functions, extreme functions are the ones that cannot be expressed as convex combinations of two distinct valid functions. In this paper we construct an extreme function $π:\mathbb{R} \to [0,1]$ whose graph is dense in $\mathbb{R} \times [0,1]$. Therefore, $π$ is discontinuous everywhere.

math.OC↗

arXiv2017

Mixed-integer linear representability, disjunctions, and Chvatal functions --- modeling implications

Jeroslow and Lowe gave an exact geometric characterization of subsets of $\mathbb{R}^n$ that are projections of mixed-integer linear sets, also known as MILP-representable or MILP-R sets. We give an alternate algebraic characterization by showing that a set is MILP-R {\em if and only if} the set can be described as the intersection of finitely many {\em affine Chvatal inequalities} in continuous variables (termed AC sets). These inequalities are a modification of a concept introduced by Blair and Jeroslow. Unlike the case for linear inequalities, allowing for integer variables in Chvatal inequalities and projection does not enhance modeling power. We show that the MILP-R sets are still precisely those sets that are modeled as affine Chvatal inequalites with integer variables. Furthermore, the projection of a set defined by affine Chvatal inequalites with integer variables is still an MILP-R set. We give a sequential variable elimination scheme that, when applied to a MILP-R set yields the AC set characterization. This is related to the elimination scheme of Williams and Williams-Hooker, who describe projections of integer sets using \emph{disjunctions} of affine Chvatal systems. We show that disjunctions are unnecessary by showing how to find the affine Chvatal inequalities that cannot be discovered by the Williams-Hooker scheme. This allows us to answer a long-standing open question due to Ryan (1991) on designing an elimination scheme to represent finitely-generated integral monoids as a system of Chvatal inequalities \emph{without} disjunctions. Finally, our work can be seen as a generalization of the approach of Blair and Jeroslow, and Schrijver for constructing consistency testers for integer programs to general AC sets.

math.OC↗

arXiv2017

Lower bounds over Boolean inputs for deep neural networks with ReLU gates

Motivated by the resurgence of neural networks in being able to solve complex learning tasks we undertake a study of high depth networks using ReLU gates which implement the function $x \mapsto \max\{0,x\}$. We try to understand the role of depth in such neural networks by showing size lowerbounds against such network architectures in parameter regimes hitherto unexplored. In particular we show the following two main results about neural nets computing Boolean functions of input dimension $n$, 1. We use the method of random restrictions to show almost linear, $Ω(ε^{2(1-δ)}n^{1-δ})$, lower bound for completely weight unrestricted LTF-of-ReLU circuits to match the Andreev function on at least $\frac{1}{2} +ε$ fraction of the inputs for $ε> \sqrt{2\frac{\log^{\frac {2}{2-δ}}(n)}{n}}$ for any $δ\in (0,\frac 1 2)$ 2. We use the method of sign-rank to show exponential in dimension lower bounds for ReLU circuits ending in a LTF gate and of depths upto $O(n^ξ)$ with $ξ< \frac{1}{8}$ with some restrictions on the weights in the bottom most layer. All other weights in these circuits are kept unrestricted. This in turns also implies the same lowerbounds for LTF circuits with the same architecture and the same weight restrictions on their bottom most layer. Along the way we also show that there exists a $\mathbb{R}^ n\rightarrow \mathbb{R}$ Sum-of-ReLU-of-ReLU function which Sum-of-ReLU neural nets can never represent no matter how large they are allowed to be.

cs.CC↗

arXiv2017

Sparse Coding and Autoencoders

In "Dictionary Learning" one tries to recover incoherent matrices $A^* \in \mathbb{R}^{n \times h}$ (typically overcomplete and whose columns are assumed to be normalized) and sparse vectors $x^* \in \mathbb{R}^h$ with a small support of size $h^p$ for some $0 <p < 1$ while having access to observations $y \in \mathbb{R}^n$ where $y = A^*x^*$. In this work we undertake a rigorous analysis of whether gradient descent on the squared loss of an autoencoder can solve the dictionary learning problem. The "Autoencoder" architecture we consider is a $\mathbb{R}^n \rightarrow \mathbb{R}^n$ mapping with a single ReLU activation layer of size $h$. Under very mild distributional assumptions on $x^*$, we prove that the norm of the expected gradient of the standard squared loss function is asymptotically (in sparse code dimension) negligible for all points in a small neighborhood of $A^*$. This is supported with experimental evidence using synthetic data. We also conduct experiments to suggest that $A^*$ is a local minimum. Along the way we prove that a layer of ReLU gates can be set up to automatically recover the support of the sparse codes. This property holds independent of the loss function. We believe that it could be of independent interest.

cs.LG↗

arXiv2017

Minimal cut-generating functions are nearly extreme

We study continuous (strongly) minimal cut generating functions for the model where all variables are integer. We consider both the original Gomory-Johnson setting as well as a recent extension by Cornuéjols and Yıldız. We show that for any continuous minimal or strongly minimal cut generating function, there exists an extreme cut generating function that approximates the (strongly) minimal function as closely as desired. In other words, the extreme functions are "dense" in the set of continuous (strongly) minimal functions.

math.OC↗

arXiv2017

Extreme functions with an arbitrary number of slopes

For the one dimensional infinite group relaxation, we construct a sequence of extreme valid functions that are piecewise linear and such that for every natural number $k\geq 2$, there is a function in the sequence with $k$ slopes. This settles an open question in this area regarding a universal bound on the number of slopes for extreme functions. The function which is the pointwise limit of this sequence is an extreme valid function that is continuous and has an infinite number of slopes. This provides a new and more refined counterexample to an old conjecture of Gomory and Johnson stating that all extreme functions are piecewise linear. These constructions are extended to obtain functions for the higher dimensional group problems via the sequential-merge operation of Dey and Richard.

math.OC↗

arXiv2017

Approximation of Minimal Functions by Extreme Functions

In a recent paper, Basu, Hildebrand, and Molinaro established that the set of continuous minimal functions for the 1-dimensional Gomory-Johnson infinite group relaxation possesses a dense subset of extreme functions. The $n$-dimensional version of this result was left as an open question. In the present paper, we settle this query in the affirmative: for any integer $n \geq 1$, every continuous minimal function can be arbitrarily well approximated by an extreme function in the $n$-dimensional Gomory-Johnson model.

math.OC↗

arXiv2017

Approximation of corner polyhedra with families of intersection cuts

We study the problem of approximating the corner polyhedron using intersection cuts derived from families of lattice-free sets in $\mathbb{R}^n$. In particular, we look at the problem of characterizing families that approximate the corner polyhedron up to a constant factor, which depends only on $n$ and not the data or dimension of the corner polyhedron. The literature already contains several results in this direction. In this paper, we use the maximum number of facets of lattice-free sets in a family as a measure of its complexity and precisely characterize the level of complexity of a family required for constant factor approximations. As one of the main results, we show that, for each natural number $n$, a corner polyhedron with $n$ basic integer variables and an arbitrary number of continuous non-basic variables is approximated up to a constant factor by intersection cuts from lattice-free sets with at most $i$ facets if $i> 2^{n-1}$ and that no such approximation is possible if $i \leq 2^{n-1}$. When the approximation factor is allowed to depend on the denominator of the fractional vertex of the linear relaxation of the corner polyhedron, we show that the threshold is $i > n$ versus $i \leq n$. The tools introduced for proving such results are of independent interest for studying intersection cuts.

math.OC↗

arXiv2017

A geometric approach to cut-generating functions

The cutting-plane approach to integer programming was initiated more that 40 years ago: Gomory introduced the corner polyhedron as a relaxation of a mixed integer set in tableau form and Balas introduced intersection cuts for the corner polyhedron. This line of research was left dormant for several decades until relatively recently, when a paper of Andersen, Louveaux, Weismantel and Wolsey generated renewed interest in the corner polyhedron and intersection cuts. Recent developments rely on tools drawn from convex analysis, geometry and number theory, and constitute an elegant bridge between these areas and integer programming. We survey these results and highlight recent breakthroughs in this area.

math.OC↗

arXiv2017

Convex Sets and Minimal Sublinear Functions

We show that, given a closed convex set $K$ containing the origin in its interior, the support function of the set $\{y\in K^*: \exists x\in K\mbox{ such that } \langle x,y \rangle =1\}$ is the pointwise smallest among all sublinear functions $σ$ such that $K=\{x: σ(x)\leq 1\}$.

math.MG↗

arXiv2017

On the Relative Strength of Split, Triangle and Quadrilateral Cuts

Integer programs defined by two equations with two free integer variables and nonnegative continuous variables have three types of nontrivial facets: split, triangle or quadrilateral inequalities. In this paper, we compare the strength of these three families of inequalities. In particular we study how well each family approximates the integer hull. We show that, in a well defined sense, triangle inequalities provide a good approximation of the integer hull. The same statement holds for quadrilateral inequalities. On the other hand, the approximation produced by split inequalities may be arbitrarily bad.

math.OC↗

arXiv2017

A Counterexample to a Conjecture of Gomory and Johnson

In Mathematical Programming 2003, Gomory and Johnson conjecture that the facets of the infinite group problem are always generated by piecewise linear functions. In this paper we give an example showing that the Gomory-Johnson conjecture is false.

math.OC↗

arXiv2017

Unique lifting of integer variables in minimal inequalities

This paper contributes to the theory of cutting planes for mixed integer linear programs (MILPs). Minimal valid inequalities are well understood for a relaxation of an MILP in tableau form where all the nonbasic variables are continuous; they are derived using the gauge function of maximal lattice-free convex sets. In this paper we study lifting functions for the nonbasic integer variables starting from such minimal valid inequalities. We characterize precisely when the lifted coefficient is equal to the coefficient of the corresponding continuous variable in every minimal lifting. The answer is a nonconvex region that can be obtained as a finite union of convex polyhedra. We then establish a necessary and sufficient condition for the uniqueness of the lifting function.

math.OC↗

arXiv2017

Characterization of the Split Closure via Geometric Lifting

We analyze split cuts from the perspective of cut generating functions via geometric lifting. We show that $α$-cuts, a natural higher-dimensional generalization of the $k$-cuts of Cornuéjols et al., gives all the split cuts for the mixed-integer corner relaxation. As an immediate consequence we obtain that the $k$-cuts are equivalent to split cuts for the 1-row mixed-integer relaxation. Further, we show that split cuts for finite-dimensional corner relaxations are restrictions of split cuts for the infinite-dimensional relaxation. In a final application of this equivalence, we exhibit a family of pure-integer programs whose split closures have arbitrarily bad integrality gap. This complements the mixed-integer example provided by Basu et al [On the relative strength of split, triangle and quadrilateral cuts, Math. Program. 126(2):281--314, 2011].

math.OC↗

arXiv2017

Minimal inequalities for an infinite relaxation of integer programs

We show that maximal $S$-free convex sets are polyhedra when $S$ is the set of integral points in some rational polyhedron of $\mathbb{R}^n$. This result extends a theorem of Lovász characterizing maximal lattice-free convex sets. Our theorem has implications in integer programming. In particular, we show that maximal $S$-free convex sets are in one-to-one correspondence with minimal inequalities.

math.OC↗

arXiv2017

Experiments with two-row cuts from degenerate tableaux

There has been a recent interest in cutting planes generated from two or more rows of the optimal simplex tableau. One can construct examples of integer programs for which a single cutting plane generated from two rows dominates the entire split closure. Motivated by these theoretical results, we study the effect of adding a family of cutting planes generated from two rows on a set of instances from the MIPLIB library. The conclusion of whether these cuts are competitive with GMI cuts is very sensitive to the experimental setup. In particular, we consider the issue of reliability versus aggressiveness of the cut generators, an issue that is usually not addressed in the literature.

math.OC↗

arXiv2017

Maximal lattice-free convex sets in linear subspaces

We consider a model that arises in integer programming, and show that all irredundant inequalities are obtained from maximal lattice-free convex sets in an affine subspace. We also show that these sets are polyhedra. The latter result extends a theorem of Lovász characterizing maximal lattice-free convex sets in $\mathbb{R}^n$.

math.OC↗

arXiv2017

Intersection Cuts with Infinite Split Rank

We consider mixed integer linear programs where free integer variables are expressed in terms of nonnegative continuous variables. When this model only has two integer variables, Dey and Louveaux characterized the intersection cuts that have infinite split rank. We show that, for any number of integer variables, the split rank of an intersection cut generated from a bounded convex set $P$ is finite if and only if the integer points on the boundary of $P$ satisfy a certain "2-hyperplane property". The Dey-Louveaux characterization is a consequence of this more general result.

math.OC↗