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Randomized algorithms for the least-squares approximation problem. Petros Drineas Rensselaer Polytechnic Institute Computer Science Department. For papers, etc. drineas. Overview. L 2 regression algorithms Overview of well-known numerical analysis algorithms,

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## Randomized algorithms for the least-squares approximation problem

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**Randomized algorithms for the least-squares approximation**problem Petros Drineas Rensselaer Polytechnic Institute Computer Science Department For papers, etc. drineas**Overview**• L2 regression algorithms • Overview of well-known numerical analysis algorithms, • Measuring the “importance” of each constraint • A fast randomized algorithm • The column subset selection problem (CSSP) • Deterministic and Randomized Algorithms • Summary and an open problem**Model y(t) (unknown) as a linear combination of d basis**functions: A is an n x d “design matrix” (n >> d): In matrix-vector notation, Problem definition and motivation In many applications (e.g., statistical data analysis and scientific computation), one has n observations of the form:**Least-norm approximation problems**Recall a linear measurement model: In order to estimate x, solve:**Application: data analysis in science**• First application: 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! • Data analysis: Fit parameters of a biological, chemical, economical, physical (astronomical), social, internet, etc. model to experimental data.**Norms of common interest**Let y = b and define the residual: Least-squares approximation: Chebyshev or mini-max approximation: Sum of absolute residuals approximation:**Lp regression problems**We are interested in over-constrained Lp regression problems, n >> d. Typically, there is no xsuch that Ax = b. Want to find the “best” x such that Ax ≈ b. We want to sample a subset of constraints and work only on these constraints.**Projection of b on the subspace spanned by the columns of A**Pseudoinverse of A Exact solution to L2 regression 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. Write A = QR, and solve Rx = QTb. Singular Value Decomposition: Most expensive, but best if A is very ill-conditioned. Write A = UVT, in which case: xOPT = A+b = V-1UTb. Complexity is O(nd2) , but constant factors differ.**Questions …**Let p=2. Approximation algorithms: Can we approximately solve L2 regression faster than “exact” methods? Core-sets (or induced sub-problems): Can we find a small set of constraints such that solving the L2 regression on those constraints gives an approximation to the original problem?**Randomized algorithms for Lp regression**Note: Clarkson ’05 gets a (1+)-approximation for L1 regression in O*(d3.5/4) time. He preprocessed [A,b] to make it “well-rounded” or “well-conditioned” and then sampled.**Algorithm 1: Sampling for L2 regression**• Algorithm • Fix a set of probabilities pi, i=1…n, summing up to 1. • Pick the i-th row of A and the i-th element of b with probability • min {1, rpi}, • and rescale both by (1/min{1,rpi})1/2. • Solve the induced problem. Note:in expectation, at most r rows of A and r elements of b are kept.**scaling to account for undersampling**Sampling algorithm for L2 regression sampled “rows” of b sampled rows of A**Our results for p=2**If the pi satisfy a condition, then with probability at least 1-, (A): condition number of A The sampling complexity is**0**0 Let 1¸2¸ … ¸ be the entries of . Exact computation of the SVD takes O(min{mn2 , m2n}) time. The top k left/right singular vectors/values can be computed faster using Lanczos/Arnoldi methods. SVD: formal definition : rank of A U (V): orthogonal matrix containing the left (right) singular vectors of A. S: diagonal matrix containing the singular values of A.**Notation**U(i): i-th row of U : rank of A U: orthogonal matrix containing the left singular vectors of A.**lengths of rows of matrix of left singular vectors of A**Condition on the probabilities Thecondition that the pi must satisfy is, for some (0,1] : • Notes: • O(nd2) time suffices (to compute probabilities and to construct a core-set). • As decreases, the number of rows that we need to keep increases proportionally to 1/. • Important question: • Is O(nd2) necessary?Can we compute the pi’s, or construct a core-set, faster?**Condition on the probabilities, cont’d**• Important: Sampling process must NOT loose any rank of A. • (Since pseudoinverse will amplify that error!) • Notation: • S is an r £ n matrix that samples and rescales a small number of rows of A and elements of b. • Each row of S has exactly one non-zero element (corresponding to the selected row of A); this non-zero element is set to the “rescaling value.”**Critical observation**sample & rescale sample & rescale**Critical observation, cont’d**SUA is approx. orthogonal • The approx. orthogonality of SUA allows us to prove that: • The left singular vectors of SA are (approx.) SUA. • The singular values of SA are (approx.) equal to the singular values of A. • A corollary of the above is that SA is full rank.**What if we are allowed to keep non-uniform samples of the**rows of an orthogonal matrix (scaled appropriately)? Then, in our case (n >> d), we can prove that: (Similar arguments in Frieze, Kannan, and Vempala ’98 & ‘04, D. and Kannan ’01, D. Kannan, and Mahoney ’06, Rudelson and Virshynin ’06). Sampling rows from orthogonal matrices An old question: Given an orthogonal matrix, sample a subset of its rows and argue that the resulting matrix is almost orthogonal.**Using the Hadamard matrix for preprocessing(Ailon & Chazelle**’06 used it to develop the Fast Johnson-Lindenstrauss Transform) Let D be an n£n diagonal matrix. • Multiplication of a vector by HD is “fast”, since: • (Du) is O(n) - since D is diagonal. • (HDu) is O(n logn) – use FFT-type algorithms**O(nd logd) L2 regression**Fact 1: since Hn (the n-by-n Hadamard matrix) and Dn (an n-by-n diagonal with § 1 in the diagonal, chosen uniformly at random) are orthogonal matrices, Thus, we can work with HnDnAx – HnDnb. Let’s use our sampling approach…**O(nd logd) L2 regression**Fact 1: since Hn (the n-by-n Hadamard matrix) and Dn (an n-by-n diagonal with § 1 in the diagonal, chosen uniformly at random) are orthogonal matrices, Thus, we can work with HnDnAx – HnDnb. Let’s use our sampling approach… Fact 2: Using a Chernoff-type argument, we can prove that the lengths of all the rows of the left singular vectors of HnDnA are, with probability at least .9,**O(nd logd) L2 regression**DONE! We can perform uniform sampling in order to keep r = O(d logd/2) rows of HnDnA; our L2 regression theorem guarantees the accuracy of the approximation. Running time is O(nd logd), since we can use the fast Hadamard-Walsh transform to multiply Hn and DnA. (The dependency on logd instead of logn comes from the fact that we do not need the full product of Hn and DnA.)**Deterministic algorithms?**Is it possible to come up with efficient deterministic algorithms for such tasks? A critical component in the proof was that sub-sampling an orthogonal matrix carefully allows us to get an almost orthogonal matrix again. Can we do such tasks deterministically? This led us to thinking about the …**Column Subset Selection problem (CSSP)**Given an m-by-n matrix A, find k columns of A forming an m-by-k matrix C that minimizes the above error over all O(nk) choices for C. C+: pseudoinverse of C, easily computed via the SVD of C. (If C = U VT, then C+ = V -1 UT.) PC = CC+ is the projector matrix on the subspace spanned by the columns of C.**Column Subset Selection problem (CSSP)**Given an m-by-n matrix A, find k columns of A forming an m-by-k matrix C that minimizes the above error over all O(nk) choices for C. PC = CC+ is the projector matrix on the subspace spanned by the columns of C. Complexity of the problem? O(nkmn) trivially works; NP-hard if k grows as a function of n. (NP-hardness in Civril & Magdon-Ismail ’07)**Motivation: feature selection**n features Think of data represented by matrices Numerous modern datasets are in matrix form. We are given m objects and n features describing the objects. Aij shows the “importance” of feature j for object i. The CSSP chooses features of the data that (provably) capture the structure of the data. More formally, it selects a subset of features such that all the remaining features can be expressed as linear combinations of the “chosen ones” with a small loss in accuracy. m objects**Spectral norm**Given an m-by-n matrix A, find k columns of A forming an m-by-k matrix C such that • is minimized over all O(nk) possible choices for C. • Remarks: • PCA is the projection of A on the subspace spanned by the columns of C. • The spectral or 2-norm of an m-by-n matrix X is**A lower bound for the CSS problem**For any m-by-k matrix C consisting of at most k columns of A Ak • Remarks: • This is also true if we replace the spectral norm by the Frobenius norm. • This is a – potentially – weak lower bound.**Prior work: numerical linear algebra**• Numerical Linear Algebra algorithms for CSS • Deterministic, typically greedy approaches. • Deep connection with the Rank Revealing QR factorization. • Strongest results so far (spectral norm): in O(mn2) time some function p(k,n)**Prior work: numerical linear algebra**• Numerical Linear Algebra algorithms for CSS • Deterministic, typically greedy approaches. • Deep connection with the Rank Revealing QR factorization. • Strongest results so far (Frobenius norm): in O(nk) time**Prior work: theoretical computer science**• Theoretical Computer Science algorithms for CSS • Randomized approaches, with some failure probability. • More than k rows are picked, e.g., O(poly(k)) rows. • Very strong bounds for the Frobenius norm in low polynomial time. • Not many spectral norm bounds…**The strongest Frobenius norm bound**Given an m-by-n matrix A, there exists an O(mn2) algorithm that picks at most O( k log k / 2 ) columns of A such that with probability at least 1-10-20**Prior work in TCS**• Drineas, Mahoney, and Muthukrishnan 2005 • O(mn2) time, O(k2/2) columns • Drineas, Mahoney, and Muthukrishnan 2006 • O(mn2) time, O(k log k/2) columns • Deshpande and Vempala 2006 • O(mnk2) time and O(k2 log k/2) columns • They also prove the existence of k columns of A forming a matrix C, such that • Compare to prior best existence result:**Open problems**• Design: • Faster algorithms (next slide) • Algorithms that achieve better approximation guarantees (a hybrid approach)**Prior work spanning NLA and TCS**• Woolfe, Liberty, Rohklin, and Tygert 2007 • (also Martinsson, Rohklin, and Tygert 2006) • O(mn logk) time, k columns, same spectral norm bounds as prior work • Beautiful application of the Fast Johnson-Lindenstrauss transform of Ailon-Chazelle**A hybrid approach(Boutsidis, Mahoney, and Drineas ’08)**• Given an m-by-n matrix A (assume m ¸ n for simplicity): • (Randomized phase) Run a randomized algorithm to pick c = O(k logk) columns. • (Deterministic phase) Run a deterministic algorithm on the above columns* to pick exactly k columns of A and form an m-by-k matrix C. * Not so simple …**A hybrid approach(Boutsidis, Mahoney, and Drineas ’08)**• Given an m-by-n matrix A (assume m ¸ n for simplicity): • (Randomized phase) Run a randomized algorithm to pick c = O(k logk) columns. • (Deterministic phase) Run a deterministic algorithm on the above columns* to pick exactly k columns of A and form an m-by-k matrix C. * Not so simple … Our algorithm runs in O(mn2) and satisfies, with probability at least 1-10-20,**Comparison: Frobenius norm**Our algorithm runs in O(mn2) and satisfies, with probability at least 1-10-20, • We provide an efficient algorithmic result. • We guarantee a Frobenius norm bound that is at most (k logk)1/2 worse than the best known existential result.**Comparison: spectral norm**Our algorithm runs in O(mn2) and satisfies, with probability at least 1-10-20, • Our running time is comparable with NLA algorithms for this problem. • Our spectral norm bound grows as a function of (n-k)1/4 instead of (n-k)1/2! • Do notice that with respect to k our bound is k1/4log1/2k worse than previous work. • To the best of our knowledge, our result is the first asymptotic improvement of the work of Gu & Eisenstat 1996.**Randomized phase: O(k log k) columns**• Randomized phase: c = O(k logk) • Compute probabilities pj summing to 1 • For each j = 1,2,…,n, pick the j-th column of A with probability min{1,cpj} • Let C be the matrix consisting of the chosen columns • (C has – in expectation – at most c columns)**Subspace sampling**Vk: orthogonal matrix containing the top k right singular vectors of A. S k: diagonal matrix containing the top k singular values of A. Remark: We need more elaborate subspace sampling probabilities than previous work. Subspace sampling in O(mn2) time Normalization s.t. the pj sum up to 1**Deterministic phase: k columns**• Deterministic phase • Let S1 be the set of indices of the columns selected by the randomized phase. • Let (VkT)S1 denote the set of columns of VkT with indices in S1, • (An extra technicality is that the columns of (VkT)S1 must be rescaled …) • Run a deterministic NLA algorithm on (VkT)S1to select exactly k columns. • (Any algorithm with p(k,n) = k1/2(n-k)1/2 will do.) • Let S2 be the set of indices of the selected columns (the cardinality of S2 is exactly k). • Return AS2 (the columns of A corresponding to indices in S2) as the final output.**Feature selection with the CSSP(Boutsidis, Mahoney, Drineas**’08 Mahoney & Drineas ’08Paschou, Ziv, …, Drineas, PLoS Genet ’07 Drineas, Paschou, …, Ziv, PLoS Genet ’08) S&P 500 data: • historical stock prices for ≈500 stocks for ≈1150 days in 2003-2007 • very low rank (so good methodological test), but doesn’t classify so well in low-dim space TechTC term-document data: • benchmark term-document data from the Open Directory Project (ODP) • hundreds of matrices, each ≈200 documents from two ODP categories and ≥10K terms • sometimes classifies well in low-dim space, and sometimes not DNA SNP data from HapMap: • Single nucleotide polymorphism (i.e., genetic variation) data from HapMap • hundreds of individuals and millions of SNPs - often classifies well in low-dim space**Summary**• Relative error core-set construction for l2 regression without looking at the target vector b. • O(nd log d) relative error approximation algorithm for least-squares regression. • Novel bounds for the Column Subset Selection Problem by combining randomized and deterministic results.**Open problem**Apply the ideas of this talk to sparse approximations Problem definition: Given a set of n basis vectors in Rd, with n >> d, solve ( is an accuracy parameter; the l0 norm counts the number of non-zero elements in x.) In words, we seek to (approximately) express b as a linear combination of a minimal number of basis vectors.**Papers**• Lp regression papers (theory): • Drineas, Mahoney, & Muthukrishnan, SODA ’06 • DasGupta, Drineas, Harb, Kumar, & Mahoney, SODA ’08 • Drineas, Mahoney, Muthukrishnan, Sarlos, under review, ’08 • L2 regression for learning (applications): • DasGupta, Drineas, Harb, Josifovski, & Mahoney, KDD ’07 • CSSP (theory + applications): • Boutsidis, Mahoney, & Drineas, under review, ’08 • Mahoney & Drineas, under review, ’08

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