Faster least squares approximation
Tamas Sarlos
Yahoo! Research
IPAM 23 Oct 2007
http://www.ilab.sztaki.hu/~stamas
Joint work with P. Drineas, M. Mahoney, and S.
Muthukrishnan.
Outline
Randomized methods for
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Least squares approximation
Lp regression
Regularized regression
Low rank matrix approximation
Models, curve fitting, and data analysis
In MANY applications (in statistical data analysis and scientific
computation), one has n observations (values of a dependent
variable y measured at values of an independent variable t):
Model y(t) by a linear combination of d basis functions:
A is an n x d “design matrix” with elements:
In matrix-vector notation:
Least Squares (LS) Approximation
We are interested in over-constrained L2 regression problems, n >> d.
Typically, there is no x such that Ax = b.
Want to find the “best” x such that Ax ≈ b.
Ubiquitous in applications & central to theory:
Statistical interpretation: best linear unbiased estimator.
Geometric interpretation: orthogonally project b onto span(A).
Many applications of this!
• Astronomy: Predicting the orbit of the asteroid Ceres (in 1801!).
Gauss (1809) -- see also Legendre (1805) and Adrain (1808).
First application of “least squares optimization” and runs in O(nd2) time!
• Bioinformatics: Dimension reduction for classification of gene expression microarray data.
• Medicine: Inverse treatment planning and fast intensity-modulated radiation therapy.
• Engineering: Finite elements methods for solving Poisson, etc. equation.
• Control theory: Optimal design and control theory problems.
• Economics: Restricted maximum-likelihood estimation in econometrics.
• Image Analysis: Array signal and image processing.
• Computer Science: Computer vision, document and information retrieval.
• Internet Analysis: Filtering and de-noising of noisy internet data.
• Data analysis: Fit parameters of a biological, chemical, economic, social, internet, etc. model
to experimental data.
Exact solution to LS Approximation
Cholesky Decomposition:
If A is full rank and well-conditioned,
decompose ATA = RTR, where R is upper triangular, and
solve the normal equations: RTRx=ATb.
QR Decomposition:
Slower but numerically stable, esp. if A is rank-deficient.
Projection of b on
the subspace spanned
by the columns of A
Write A=QR, and solve Rx = QTb.
Singular Value Decomposition:
Most expensive, but best if A is very ill-conditioned.
Write A=UVT, in which case: xOPT = A+b = V-1kUTb.
Complexity is O(nd2) for all of these, but
constant factors differ.
Pseudoinverse
of A
Reduction of LS Approximation to fast
matrix multiplication
Theorem: For any n-by-d matrix A, and d dimensional vector b,
where n>d, we can compute x*=A+b in O(n/dM(d)) time. Here M(d)
denotes the time needed to multiply two d-by-d matrices.
• Current best M(d)=d, =2.376… Coppersmith–Winograd (1990)
Proof: Ibarra et al. (1982) generalize the LUP decomposition for
rectangular matrixes
Direct proof:
• Recall the normal equations ATAx = ATb
• Group the rows of A into blocks of d and compute ATA as
the sum of n/d square products, each d-by-d
• Solve the normal equations
• LS in O(nd1.376) time, but impractical. We assume O(nd2).
Original (expensive) sampling algorithm
Drineas, Mahoney, and Muthukrishnan (SODA, 2006)
Algorithm
Theorem:
1.
Let:
2.
Fix a set of probabilities pi,
i=1,…,n.
Pick the i-th row of A and the i-th
element of b with probability
If the pi satisfy:
min {1, rpi},
and rescale by (1/min{1,rpi})1/2.
3.
Solve the induced problem.
for some   (0,1], then w.p. ≥ 1-,
• These probabilities pi’s are statistical leverage scores!
• “Expensive” to compute, O(nd2) time, these pi’s!
Fast LS via uniform sampling
Drineas, Mahoney, Muthukrishnan, and Sarlos (2007)
Algorithm
1.
Pre-process A and b with a “randomized
Hadamard transform”.
2.
Uniformly sample
r=O(d log(n) log(d log(n))/)
constraints.
3.
Solve the induced problem:
• Non-uniform sampling will work, but it is “slow.”
• Uniform sampling is “fast,” but will it work?
A structural lemma
Approximate
where
Let
by solving
is any matrix.
be the matrix of left singular vectors of A.
assume that -fraction of mass of b lies in span(A).
Lemma: Assume that:
Then, we get relative-error approximation:
Randomized Hadamard preprocessing
Facts implicit or explicit in: Ailon & Chazelle (2006), or Ailon and Liberty (2008).
Let Hn be an n-by-n deterministic Hadamard matrix, and
Let Dn be an n-by-n random diagonal matrix with +1/-1 chosen u.a.r. on the diagonal.
Fact 1: Multiplication by HnDn doesn’t change the solution:
(since Hn and Dn are orthogonal matrices).
Fact 2: Multiplication by HnDn is fast - only O(n log(r)) time, where r is the number of
elements of the output vector we need to “touch”.
Fact 3: Multiplication by HnDn approximately uniformizes all leverage scores:
A matrix perturbation theorem
Establishing structural conditions
Lemma:
|1-i2(SHUA)| ≤ 1/10, with constant probability,
i.e,. singular values of SHUA are close to 1.
Proof: Apply spectral norm matrix perturbation bound.
Lemma:
||(SHUA)TSHb|| ≤  OPT, with constant probability,
i.e., SHUA is nearly orthogonal to b.
Proof: Apply Forbenius norm matrix perturbation bound.
Bottom line:
• Fast algorithm approximately uniformizes the leverage of each data point.
• Uniform sampling is optimal up to O(1/log(n)) factor, so over-sample a bit.
• Overall running time O(ndlog(…)+d3log(…)/)
Fast LS via sparse projection
Drineas, Mahoney, Muthukrishnan, and Sarlos (2007) - sparse projection matrix from Matousek’s
version of Ailon-Chazelle 2006
Algorithm
1.
Pre-process A and b with a randomized Hadamard transform.
2.
Multiply preprocessed input by sparse random k x n matrix T,
where
and where k=O(d/) and q=O(d log2(n)/n+d2log(n)/n) .
3.
Solve the induced problem:
• Dense projections will work, but it is “slow.”
• Sparse projection is “fast,” but will it work?
-> YES! Sparsity parameter q of T related to non-uniformity of leverage scores!
Lp regression problems
Dasgupta, Drineas, Harb, Kumar, and Mahoney (2008)
We are interested in over-constrained Lp regression problems, n >> d.
Lp regression problems are convex programs (or better!).
There exist poly-time algorithms.
We want to solve them faster!
Lp norms and their unit balls
Recall the Lp norm for
:
Some inequality relationships include:
Solution to Lp regression
Lp regression can be cast as a convex program for all
.
For p=1, Sum of absolute residuals approximation (minimize ||Ax-b||1):
For p=∞, Chebyshev or mini-max approximation (minimize ||Ax-b||∞):
For p=2, Least-squares approximation (minimize ||Ax-b||2):
What made the L2 result work?
The L2 sampling algorithm worked because:
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For p=2, an orthogonal basis (from SVD, QR, etc.) is a “good” or “wellconditioned” basis.
(This came for free, since orthogonal bases are the obvious choice.)
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Sampling w.r.t. the “good” basis allowed us to perform “subspacepreserving sampling.”
(This allowed us to preserve the rank of the matrix and distances in
its span.)
Can we generalize these two ideas to p2?
Sufficient condition
Easy
Approximate minx||b-Ax||p by solving minx||S(b-Ax)||p giving x’
Assume that the matrix S is an isometry over the span of A and b:
for any vector y in span(Ab) we have (1-)||y||p  ||Sy||p  (1+ ) ||y||
Then, we get relative-error approximation:
||b-Ax||p  (1+ ) minx|| S(b-Ax)||p
(This requires O(d/2) for L2 regression.)
How to construct an Lp isometry?
p
p-well-conditioned basis (definition)
Let A be an n x m matrix of rank d<<n, let p  [1,∞), and q its dual.
Definition: An n x d matrix U is an (,,p)-well-conditioned basis for
span(A) if:
(1) |||U|||p ≤ , (where |||U|||p = (ij|Uij|p)1/p )
(2) for all z in Rd, ||z||q ≤  ||Uz||p.
U is a p-well-conditioned basis if ,=dO(1), independent of m,n.
p-well-conditioned basis (existence)
Let A be an n x m matrix of rank d<<n, let p  [1,∞), and q its dual.
Theorem: There exists an (,,p)-well-conditioned basis U for span(A)
s.t.:
if p < 2, then  = d1/p+1/2 and  = 1,
if p = 2, then  = d1/2 and  = 1,
if p > 2, then  = d1/p+1/2 and  = d1/q-1/2.
U can be computed in O(nmd+nd5log n) time (or just O(nmd) if p = 2).
p-well-conditioned basis (construction)
Algorithm:
• Let A=QR be any QR decomposition of A.
(Stop if p=2.)
• Define the norm on Rd by ||z||Q,p  ||Qz||p.
• Let C be the unit ball of the norm ||•||Q,p.
• Let the d x d matrix F define the Lowner-John ellipsoid of C.
• Decompose F=GTG,
where G is full rank and upper triangular.
• Return U = QG-1
as the p-well-conditioned basis.
Subspace-preserving sampling
Let A be an n x m matrix of rank d<<n, let p  [1,∞).
Let U be an (,,p)-well-conditioned basis for span(A),
Theorem: Randomly sample rows of A according to the probability
distribution:
where:
Then, with probability 1- , the following holds for all x in Rm:
Regularized regression
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If you don’t trust your data (entirely)
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Minimize ||b-Ax||p+ ||x||r, where r arbitrary
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Lasso is especially popular: minx||b-Ax||2+ ||x||1
•
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Sampled Lp regression works for the regularized
version too
We want more: imagine 
Ridge regression
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Tikhonov regularization, back to L2
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Minimize ||b-Ax||22+ 2||x||22
•
Equivalent to augmenting n-by-d A with d and b with 0
•
SVD of A UDiag(sqrt(i))VT
•
Let D = i sqrt(i) << d
•
For (1+) relative error:
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precondition with randomized FFT
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sample r=O(D log(n) log(D log(n))/) rows uniformly.
SVD of a matrix
Any m x n matrix A can be decomposed as:
U (V): orthogonal matrix containing the left (right) singular vectors of A.
: diagonal matrix containing the singular values of A, ordered non-increasingly.
: rank of A, the number of non-zero singular values.
Exact computation of the SVD takes O(min{mn2 , m2n}) time.
SVD and low-rank approximations
Truncate the SVD by keeping k ≤  terms:
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.
• Ak is the “best” matrix among all rank-k matrices wrt. to the spectral and Frobenius
norms
• This optimality property of very useful in, e.g., Principal Component Analysis (PCA),
LSI, etc.
Approximate SVD
• Starting with the seminal work of Frieze et al (1998)
and Papadimitriou et al (1998) till 2005 several results
of the form
||A-Ak’||F ≤ ||A-Ak||F + t||A||F
• ||A||F might be a significantly larger than ||A-Ak||F
• Four independent results [H-P, DV, MRT, S 06] on
||A-Ak’||F ≤ (1+)||A-Ak||F
• This year Shyanalkumar and Deshpande …
Relative error SVD via
random projections
Theorem Let S be n by r=O(k/) random matrix with
1 entries. Project A to the subspace spanned by the
rows of SA and then compute Ak’ as
the rank-k SVD of the projection. Then
||A-Ak’||F ≤ (1+)||A-Ak||F with prob. at least 1/3
• By keeping an orthonormal basis of SA total time is
O(Mr+(n+m)r^2) with M non-zeroes
• We need to make only 2 passes over the data
Can use fast random projections and avoid the SA
projection, spectral norm (talks by R and T)
SVD proof sketch
Can show:
||A-Ak’||F2 ≤ ||A-Ak||F2 + ||Ak-Ak(SAk)+(SA)||F2
For all j columns of A consider the regression A(j)≈Akx
Exact: Ak(j), error is ||A(j)-Ak(j)||2
Approximate: Ak-Ak(SAk)+SA(j)
Conclusion
Fast methods based on randomized
preprocessing and sampling or sparse
projections for
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Least squares approximation
Regularized regression
Low rank matrix approximation
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Thank you!