This paper demonstrates theoretically and empirically that a greedy algorithm called orthogonal matching pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension d given O(m ln d) random linear measurements of that signal. This is a massive improvement over previous results, which require O(m2) measurements. The new results for OMP are comparable with recent results for another approach called basis pursuit (BP). In some settings, the OMP algorithm is faster and easier to implement, so it is an attractive alternative to BP for signal recovery problems.

/pdf/signal-recovery-from-random-measurements-via-orthogonal-5adhd8dwmq.pdf

Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit

This report demonstrates theoretically and empirically that a greedy algorithm called
Orthogonal Matching Pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension
d given O(mln d) random linear measurements of that signal. This is a massive improvement
over previous results, which require O(m2) measurements. The new results for OMP are comparable
with recent results for another approach called Basis Pursuit (BP). In some settings, the
OMP algorithm is faster and easier to implement, so it is an attractive alternative to BP for signal
recovery problems.

/pdf/signal-recovery-from-random-measurements-via-orthogonal-2uks7d8txo.pdf

Signal Recovery from Random Measurements Via Orthogonal Matching Pursuit: The Gaussian Case

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

The use of randomness is now an accepted tool in Theoretical Computer Science but not everyone is aware of the underpinnings of this methodology in Combinatorics - particularly, in what is now called the probabilistic Method as developed primarily by Paul Erdoős over the past half century. Here I will explore a particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets. A central point will be the evolution of these problems from the purely existential proofs of Erdős to the algorithmic aspects of much interest to this audience.

/pdf/the-probabilistic-method-5g63k0itke.pdf

The Probabilistic Method

We present two algorithms for the approximate nearest neighbor problem in high-dimensional spaces. For data sets of size n living in R d , the algorithms require space that is only polynomial in n and d, while achieving query times that are sub-linear in n and polynomial in d. We also show applications to other high-dimensional geometric problems, such as the approximate minimum spanning tree. The article is based on the material from the authors' STOC'98 and FOCS'01 papers. It unifies, generalizes and simplifies the results from those papers.

/pdf/approximate-nearest-neighbors-towards-removing-the-curse-of-27lmlz5prr.pdf

Approximate nearest neighbors: towards removing the curse of dimensionality

: A crucial lemma in recent work of the author (showing that k-term arithmetic progression-free sets of integers must have density zero) stated (approximately) that any large bipartite graph can be decomposed into relatively few 'nearly regular' bipartite subgraphs. In this note the author generalizes this result to arbitrary graphs, at the same time strengthening and simplifying the original bipartite result.

Regular partitions of graphs

In 1926 van der Waerden [13] proved the following startling theorem : If the set of integers is arbitrarily partitioned into two classes then at least one class contains arbitrarily long arithmetic progressions. It is well known and obvious that neither class must contain an infinite arithmetic progression. In fact, it is easy to see that for any sequence an there is another sequence bn9 with bn > an9 which contains no arithmetic progression of three terms, but which intersects every infinite arithmetic progression. The finite form of van der Waerden's theorem goes as follows: For each positive integer n9 there exists a least integer f{n) with the property that if the integers from 1 to /(/?) are arbitrarily partitioned into two classes, then at least one class contains an arithmetic progression of « terms. (For a short proof, see the note of Graham and Rothschild [5].) However, the best upper bound on f{n) known at present is extremely poor. The best lower bound known, due to Berlekamp [3], asserts that/(«) < nl9 for n prime, which improves previous results of Erdös, Rado and W. Schmidt. More than 40 years ago, Erdös and Turân [4] considered the quantity rk{n)9 defined to be the greatest integer / for which there is a sequence of integers 0 < a\\ < a2 < ••• < a; ^ n which does not contain an arithmetic progression of k terms. They were led to the investigation of rk{n) by several things. First of all the problem of estimating rk{n) is clearly interesting in itself. Secondly, rk{n) < n/2 would imply f{k) < 77, i.e., they hoped to improve the poor upper bound on f{k) by investigating rk{n). Finally, an old question in number theory asks if there are arbitrarily long arithmetic progressions of prime numbers. From rk{n) < %{rì) this would follow immediately. The hope was that this problem on primes could be attacked not by

/pdf/on-sets-of-integers-containing-k-elements-in-arithmetic-1mfhvbcik7.pdf

On sets of integers containing k elements in arithmetic progression

A data structure for representing a set of n items from a umverse of m items, which uses space n + o(n) and accommodates membership queries m constant time is described. Both the data structure and the query algorithm are easy to ~mplement.

/pdf/storing-a-sparse-table-with-0-1-worst-case-access-time-1a5jde8jx8.pdf

Storing a Sparse Table with 0(1) Worst Case Access Time

The purpose of this paper is to describe a sorting network of size 0(n log n) and depth 0(log n). A natural way of sorting is through consecutive halvings: determine the upper and lower halves of the set, proceed similarly within the halves, and so on. Unfortunately, while one can halve a set using only 0(n) comparisons, this cannot be done in less than log n (parallel) time, and it is known that a halving network needs (½)n log n comparisons. It is possible, however, to construct a network of 0(n) comparisons which halves in constant time with high accuracy. This procedure (e-halving) and a derived procedure (e-nearsort) are described below, and our sorting network will be centered around these elementary steps.

An 0(n log n) sorting network

In this paper, we establish several theorems involving configurations of points and lines in the Euclidean plane. Our results answer questions and settle conjectures of P. Erdos, G. Purdy, and G. Dirac. The principal result is that there exists an absolute constantc
 1 so that when
 
$$\sqrt n \leqq t \leqq \left( {_2^n } \right)$$

 , the number of incidences betweenn points andt lines is less thanc
 1
 n
 2/3
 t
 2/3. Using this result, it follows immediately that there exists an absolute constantc
 2 so that ifk≦√n, then the number of lines containing at leastk points is less thanc
 2
 n
 2/k
 3. We then prove that there exists an absolute constantc
 3 so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc
 3
 n of the lines determined by then points. Finally, we show that there is an absolute constantc
 4 so that there are less than exp (c
 4 √n) sequences 2≦y
 1≦y
 2≦...≦y
 r for which there is a set ofn points and a setl
 1,l
 2, ...,l
 t
 oft lines so thatl
 j
 containsy
 j
 points.

/pdf/extremal-problems-in-discrete-geometry-2l1usdjbzz.pdf

Endre Szemerédi

Papers

Regular partitions of graphs

On sets of integers containing k elements in arithmetic progression

Storing a Sparse Table with 0(1) Worst Case Access Time

An 0(n log n) sorting network

Extremal problems in discrete geometry