arXiv subjects

Boris Alexeev

Publications and source records attributed to Boris Alexeev.

29 records · Page 2Linked to original sources

arXiv2013

Phase retrieval with polarization

In many areas of imaging science, it is difficult to measure the phase of linear measurements. As such, one often wishes to reconstruct a signal from intensity measurements, that is, perform phase retrieval. In this paper, we provide a novel measurement design which is inspired by interferometry and exploits certain properties of expander graphs. We also give an efficient phase retrieval procedure, and use recent results in spectral graph theory to produce a stable performance guarantee which rivals the guarantee for PhaseLift in [Candes et al. 2011]. We use numerical simulations to illustrate the performance of our phase retrieval procedure, and we compare reconstruction error and runtime with a common alternating-projections-type procedure.

cs.IT↗

arXiv2012

Full Spark Frames

Finite frame theory has a number of real-world applications. In applications like sparse signal processing, data transmission with robustness to erasures, and reconstruction without phase, there is a pressing need for deterministic constructions of frames with the following property: every size-M subcollection of the M-dimensional frame elements is a spanning set. Such frames are called full spark frames, and this paper provides new constructions using the discrete Fourier transform. Later, we prove that full spark Parseval frames are dense in the entire set of Parseval frames, meaning full spark frames are abundant, even if one imposes an additional tightness constraint. Finally, we prove that testing whether a given matrix is full spark is hard for NP under randomized polynomial-time reductions, indicating that deterministic full spark constructions are particularly significant because they guarantee a property which is otherwise difficult to check.

math.FA↗

arXiv2012

A rearrangement step with potential uses in priority queues

Link-based data structures, such as linked lists and binary search trees, have many well-known rearrangement steps allowing for efficient implementations of insertion, deletion, and other operations. We describe a rearrangement primitive designed for link-based, heap-ordered priority queues in the comparison model, such as those similar to Fibonacci heaps or binomial heaps. In its most basic form, the primitive rearranges a collection of heap-ordered perfect binary trees. Doing so offers a data structure control on the number of trees involved in such a collection, in particular keeping this number logarithmic in the number of elements. The rearrangement step is free from an amortized complexity standpoint (using an appropriate potential function).

cs.DS↗

arXiv2011

Tensor Rank: Some Lower and Upper Bounds

The results of Strassen and Raz show that good enough tensor rank lower bounds have implications for algebraic circuit/formula lower bounds. We explore tensor rank lower and upper bounds, focusing on explicit tensors. For odd d, we construct field-independent explicit 0/1 tensors T:[n]^d->F with rank at least 2n^(floor(d/2))+n-Theta(d log n). This matches (over F_2) or improves (all other fields) known lower bounds for d=3 and improves (over any field) for odd d>3. We also explore a generalization of permutation matrices, which we denote permutation tensors. We show, by counting, that there exists an order-3 permutation tensor with super-linear rank. We also explore a natural class of permutation tensors, which we call group tensors. For any group G, we define the group tensor T_G^d:G^d->F, by T_G^d(g_1,...,g_d)=1 iff g_1 x ... x g_d=1_G. We give two upper bounds for the rank of these tensors. The first uses representation theory and works over large fields F, showing (among other things) that rank_F(T_G^d)<= |G|^(d/2). We also show that if this upper bound is tight, then super-linear tensor rank lower bounds would follow. The second upper bound uses interpolation and only works for abelian G, showing that over any field F that rank_F(T_G^d)<= O(|G|^(1+log d)log^(d-1)|G|). In either case, this shows that many permutation tensors have far from maximal rank, which is very different from the matrix case and thus eliminates many natural candidates for high tensor rank. We also explore monotone tensor rank. We give explicit 0/1 tensors T:[n]^d->F that have tensor rank at most dn but have monotone tensor rank exactly n^(d-1). This is a nearly optimal separation.

cs.CC↗

arXiv2010

Forbidden induced subgraphs of double-split graphs

In the course of proving the strong perfect graph theorem, Chudnovsky, Robertson, Seymour, and Thomas showed that every perfect graph either belongs to one of five basic classes or admits one of several decompositions. Four of the basic classes are closed under taking induced subgraphs (and have known forbidden subgraph characterizations), while the fifth one, consisting of double-split graphs, is not. A graph is doubled if it is an induced subgraph of a double-split graph. We find the forbidden induced subgraph characterization of doubled graphs; it contains 44 graphs.

math.CO↗

arXiv2010

On minimal colorings without monochromatic solutions to a linear equation

For a ring R and system L of linear homogeneous equations, we call a coloring of the nonzero elements of R minimal for L if there are no monochromatic solutions to L and the coloring uses as few colors as possible. For a rational number q and positive integer n, let E(q,n) denote the equation $\sum_{i=0}^{n-2} q^{i}x_i = q^{n-1}x_{n-1}$. We classify the minimal colorings of the nonzero rational numbers for each of the equations E(q,3) with q in {3/2,2,3,4}, for E(2,n) with n in {3,4,5,6}, and for x_1+x_2+x_3=4x_4. These results lead to several open problems and conjectures on minimal colorings.

math.CO↗

arXiv2010

On the complexity of Mumford-Shah type regularization, viewed as a relaxed sparsity constraint

We show that inverse problems with a truncated quadratic regularization are NP-hard in general to solve, or even approximate up to an additive error. This stands in contrast to the case corresponding to a finite-dimensional approximation to the Mumford-Shah functional, where the operator involved is the identity and for which polynomial-time solutions are known. Consequently, we confirm the infeasibility of any natural extension of the Mumford-Shah functional to general inverse problems. A connection between truncated quadratic minimization and sparsity-constrained minimization is also discussed.

math.OC↗

arXiv2010

An analysis of a war-like card game

In his book "Mathematical Mind-Benders", Peter Winkler poses the following open problem, originally due to the first author: "[In the game Peer Pressure,] two players are dealt some number of cards, initially face up, each card carrying a different integer. In each round, the players simultaneously play a card; the higher card is discarded and the lower card passed to the other player. The player who runs out of cards loses. As the number of cards dealt becomes larger, what is the limiting probability that one of the players will have a winning strategy?" We show that the answer to this question is zero, as Winkler suspected. Moreover, assume the cards are dealt so that one player receives r >= 1 cards for every one card of the other. Then if r < phi = (1+sqrt 5)/2, the limiting probability that either player has a winning strategy is still zero, while if r > phi, it is one.

math.CO↗

arXiv2009

Equations resolving a conjecture of Rado on partition regularity

A linear equation L is called k-regular if every k-coloring of the positive integers contains a monochromatic solution to L. Richard Rado conjectured that for every positive integer k, there exists a linear equation that is (k-1)-regular but not k-regular. We prove this conjecture by showing that the equation $\sum_{i=1}^{k-1} \frac{2^i}{2^i-1} x_i = (-1 + \sum_{i=1}^{k-1} \frac{2^i}{2^i-1}) x_0$ has this property. This conjecture is part of problem E14 in Richard K. Guy's book "Unsolved problems in number theory", where it is attributed to Rado's 1933 thesis, "Studien zur Kombinatorik".

math.CO↗

arXiv2006

On lengths of rainbow cycles

We prove several results regarding edge-colored complete graphs and rainbow cycles, cycles with no color appearing on more than one edge. We settle a question posed by Ball, Pultr, and Vojtěchovský by showing that if such a coloring does not contain a rainbow cycle of length $n$, where $n$ is odd, then it also does not contain a rainbow cycle of length $m$ for all $m$ greater than $2n^2$. In addition, we present two examples which demonstrate that this result does not hold for even $n$. Finally, we state several open problems in the area.

math.CO↗

arXiv2003

Minimal DFAs for Testing Divisibility

We present and prove a theorem answering the question "how many states does a minimal deterministic finite automaton (DFA) that recognizes the set of base-b numbers divisible by k have?"

cs.CC↗