of Technology Lecturer : Piotr Indyk

Massachusetts Institute of Technology Lecturer : Piotr Indyk

Let x be a vector variable taking values from Zm. We consider a setting where the value of x changes over time through updates of its coordinate values. Every update can be specified in form of a pair (i, a), 1 ≤ i ≤ m and a is an integer, which has a meaning of increasing the value of xi by a. A sequence of such pairs is called a stream in our context. The value of x before the first update is...

Massachusetts Institute of Technology Lecturer : Piotr Indyk 6 . 895 : Sketching , Streaming and Sub - linear Space Algorithms

In previous lectures, we have seen streaming algorithms that operate a number of different data types, including numerical, metric, and geometric. Today we will investigate streaming algorithms on graphs. Our model for graph data will be as follows. Given a graph G = (V,E) with |V | = n, we will assume that V is known, and that the edges in E are revealed in arbitrary order (deletions are not s...

Massachusetts Institute of Technology Lecturer : Piotr Indyk

The goal is to acquire signals in R that are well approximated by sparse signals with k nonzero components, where k << n. The measurement process can be represented by an m× n matrix A, where m is roughly proportional to k rather than n. The recovery algorithm uses the sketch and a description of the measurement matrix to construct a signal approximation x̂ that has only O(k) nonzero components....

of Technology Lecturer : Piotr Indyk 6 . 895 : Sketching , Streaming and Sub - linear Space Algorithms

consider the case where x is has exactly k + 1 nonzero entries. Then, Errk(x) = Err 2 k(x) = x (k+1), where x(k+1) represents the smallest of the k + 1 nonzero entries in x. Thus, the formula above implies that for such x, the LP finds an x∗ that is better than the best k-sparse approximation, so clearly x∗ cannot be k-sparse. In practice, it is often not important that x∗ be k-sparse. For exam...

8 Low - Distortion Embeddings of Finitemetric