Computing Sketches of Matrices Computing Sketches of Matrices Efficiently & (Privacy Preserving) Efficiently & (Privacy Preserving) Data Mining Data Mining Petros Drineas Petros Drineas Rensselaer Polytechnic Institute [email protected](joint work with R. Kannan and M. Mahoney) @ DIMACS Workshop on Privacy Preserving Data Mining
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Computing Sketches of Matrices Efficiently & (Privacy Preserving) Data Mining Petros Drineas Rensselaer Polytechnic Institute [email protected] (joint.
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Computing Sketches of MatricesComputing Sketches of MatricesEfficiently & (Privacy Preserving) Data Efficiently & (Privacy Preserving) Data
@ DIMACS Workshop on Privacy Preserving Data Mining
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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”.
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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 !
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Our approach & our results
1. 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]
2. We draw the rows/columns randomly, using adaptive sampling; e.g. rows/columns are picked with probability proportional to their lengths.
Create an approximation to the original matrix which can be stored in much less space.
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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
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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).
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Applications (cont’d)
Key observation: The exact value xT· d might not be necessary.
1. The feature values in the vectors are set by coarse heuristics.
2. It is in general enough to see if xT· d > Threshold.
feature 1
fea
ture
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Object x
Object d
(d,x)
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.
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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.
The CUR algorithm guarantees a bound on the worst case choice of x.
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Approximating A efficiently
Given a large m-by-n matrix A (stored on disk), compute an approximation A’ to A such that:
1. 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.
2. A’ satisfies (with high probability)
||A-A’||22 < ε ||A||F
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(and a similar bound with respect to the Frobenius norm).
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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.
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Creating C and R
• 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]
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Singular Value Decomposition (SVD)
1. Exact computation of the SVD takes O(min(mn2 , m2n)) time.
2. The top few singular vectors/values can be approximated faster (Lanczos/ Arnoldi methods).
U (V): orthogonal matrix containing the left (right) singular vectors of A.
: diagonal matrix containing the singular values of A.
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Rank k approximations (Ak)
Ak is a matrix of rank k such that ||A-Ak||2,F is minimized over all rank k matrices!
Uk (Vk): orthogonal matrix containing the top k left (right) singular vectors of A.
k: diagonal matrix containing the top k singular values of A.
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The CUR algorithm
Input:
1. The matrix A in “sparse unordered representation”.
(e.g. non-zero entries of A are presented as triples (i,j,Aij) in any order)
2. Positive integers c < n and r < m (number or columns/rows that we pick).
3. 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.
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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.
e.g.
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.
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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.
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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.
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Error bounds (2-norm)
Assume Ak is the “best” rank k approximation to A (through SVD). Then
since |A-Ak|22 <= |A|F
2/(k+1).
We need to pick O(1/ε2) rows and O(1/ε2) columns.
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Can we do better?
Lemma
For any < 1, there is a set of Ω(–n) n-by-n matrices, such that for two distinct matrices A,B in the set,
||A-B||22 > (/20)||A||F
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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/)) bits
Tighter lower bounds, matching almost exactly with our upper bounds, have been obtained by Ziv-Bar Yossef, STOC ’03.
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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.
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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?