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Computing Sketches of Matrices Efficiently & (Privacy Preserving) Data Mining

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## Computing Sketches of Matrices Efficiently & (Privacy Preserving) Data Mining

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**Computing Sketches of MatricesEfficiently & (Privacy**Preserving) Data Mining Petros Drineas Rensselaer Polytechnic Institute drinep@cs.rpi.edu (joint work with R. Kannan and M. Mahoney) @ DIMACS Workshop on Privacy Preserving Data Mining**Motivation (Data Mining)**• In many applications large matrices appear (too large to store in RAM). • We can make a few “passes” (sequential READS) through the matrices. • We can create and store a small “sketch” of the matrices in RAM. • Computing the “sketch” should be a very fast process. • Discard the original matrix and work with the “sketch”.**Motivation (Privacy Preserving)**• In many applications, instead of revealing a large matrix, we only reveal its “sketch”. • Intuition: The “sketch” is an approximation to the original matrix. • Instead of viewing the approximation as a “necessary evil”, we might be able to use it to achieve privacy preservation (similar ideas in Feigenbaum et. al., ICALP 2001). • Goal: Formulate a technical definition of privacy that might be achievable by such “sketching” algorithms and provide meaningful and quantifiable protection. • Achieving the goal is an open problem !**Create an approximation to the original matrixwhich can be**stored in much less space. Our approach & our results • A “sketch” consisting of a few rows/columns of the matrix is adequate for efficient approximations. • [see D & Kannan ’03, and D, Kannan & Mahoney ’04] • We draw the rows/columns randomly, using adaptive sampling; e.g. rows/columns are picked with probability proportional to their lengths.**Overview**• A Data Mining setup • Approximating a large matrix • Algorithm • Error bounds • Tightness of the results • An alternative approach (Achlioptas and McSherry ’01 and ’03) • Conclusions**Applications: Data Mining**We are given m (>106) objects and n(>105) features describing the objects. Database An m-by-n matrix A (Aij shows the “importance” of feature j for object i). Every row of A represents an object. Queries Given a new object x, find similar objects in the database (nearest neighbors).**Applications (cont’d)**Two objects are “close” if the angle between their corresponding vectors is small. So, assuming that the vectors are normalized, xT·d = cos(x,d) is high when the two objects are close. A·x computes all the angles and answers the query. • Key observation: The exact value xT· d might not be necessary. • The feature values in the vectors are set by coarse heuristics. • It is in general enough to see if xT·d > Threshold.**The CUR algorithm guarantees a bound on the worst case**choice of x. Using an approximation to A Assume that A’ = CUR is an approximation to A, such that A’ is stored efficiently (e.g. in RAM). Given a query vector x, instead of computing A · x, compute A’ · x to identify its nearest neighbors.**Approximating A efficiently**• Given a large m-by-n matrix A (stored on disk), compute an approximation A’ to A such that: • A’ can be stored in O(m+n) space, after making two passes through the entire matrix A, and using O(m+n) additional space and time. • A’ satisfies (with high probability) • ||A-A’||22 < ε ||A||F2 • (and a similar bound with respect to the Frobenius norm).**Describing A’ = C · U · R**• C consists of c = θ(1/ε2) columns of A and R consists of r = θ(1/ε2) rows of A (the “description length” of A is O(m+n)). • C and R are created using adaptive sampling.**Create C (R) by performing c (r) i.i.d trials.**• In each trial, pick a column (row) of A with probability • Include A(i) (A(i)) as a column of C (R). • [A(i) (A(i)) is the i-th column (row) of A] Creating C and R**Singular Value Decomposition (SVD)**U (V): orthogonal matrix containing the left (right) singular vectors of A. S: diagonal matrix containing the singular values of A. • Exact computation of the SVD takes O(min(mn2 , m2n)) time. • The top few singular vectors/values can be approximated faster (Lanczos/ Arnoldi methods).**Rank k approximations (Ak)**Uk (Vk): orthogonal matrix containing the top k left (right) singular vectors of A. Sk: diagonal matrix containing the top k singular values of A. Ak is a matrix of rank k such that ||A-Ak||2,F is minimized over all rank k matrices!**The CUR algorithm**• Input: • The matrix A in “sparse unordered representation”. • (e.g. non-zero entries of A are presented as triples (i,j,Aij) in any order) • Positive integers c < n and r < m (number or columns/rows that we pick). • Positive integer k (the rank of A’=CUR). Note: Since A’ is of rank k, ||A-A’||2,F >= ||A-Ak||2,F. We choose a k such that ||A-Ak||2,F is small. As k grows, for the Frobenius norm approximation, c and r grow as well.**e.g.**Computing U • Intuition: • The CUR algorithm essentially expresses every row of the matrix A as a linear combination of a small subset of the rows of A. • This small subset consists of the rows in R. • Given a row of A – say A(i) – the algorithm computes the “best fit” for the row A(i) using the rows in R as the basis. Notice that only c = O(1) element of the i-th row are given as input. However, a vector of coefficients u can still be computed.**Creating U**Running time Computing the elements of U amounts to a pseudo-inverse computation. It can be done in O(c2m + c3 + r3) time. Thus, U can be computed in O(m) time. Note on the rank of U and CUR The rank of U (by construction) is k. Thus, the rank of A’=CUR is at most k.**Error bounds (Frobenius norm)**Assume Ak is the “best” rank k approximation to A (through SVD). Then We need to pick O(k/ε2) rows and O(k/ε2) columns.**Error bounds (2-norm)**Assume Ak is the “best” rank k approximation to A (through SVD). Then since |A-Ak|22 <= |A|F2/(k+1). We need to pick O(1/ε2) rows and O(1/ε2) columns.**Can we do better?**Lemma For any e < 1, there is a set of Ω(e–n) n-by-n matrices, such that for two distinct matrices A,B in the set, ||A-B||22 > (e/20)||A||F2 Lower bound Theorem Any algorithm which approximates these matrices must output a different “sketch” for each one, thus it must output at least Ω(n log(1/e)) bits Tighter lower bounds, matching almost exactly with our upper bounds, have been obtained by Ziv-Bar Yossef, STOC ’03.**A different technique**• (D. Achlioptas and F. McSherry, ’01 and ’03) • The Algorithm in 2 lines: • To approximate a matrix A, keep a few elements of the matrix (instead of rows or columns) and zero out the remaining elements. • Compute a rank k approximation to this sparse matrix (using Lanczos methods). • Comparing the two techniques: • The error bound w.r.t. the 2-norm is better, while the error bound w.r.t. the Frobenius norm is the same. • (weighted sampling is used - heavier elements are kept with higher probabilities) • Running times are the same.**Conclusions**• Given the small “sketch” of a matrix A, a “friendly user” can • reconstruct a (provably accurate) approximation A’ to the original matrix A and employ any algorithms that he would use to process the original matrix A on A’, • use the Frobenius and spectral norm bounds for A-A’ to argue about the approximation error of his algorithms. • How do we ensure privacy for the object-vectors (rows) of A that are revealed as part of R? • Are such sketches offering some privacy preserving guarantees, under some (relaxed) definition of privacy?