TUTORIAL: Randomized Algorithms for Matrices and Massive Data Sets

TUTORIAL: Randomized Algorithms for Matrices and Massive Data Sets
Petros Drineas RPI Dept. of Computer Science Michael W. Mahoney Yale University Dept. of Mathematics (now at Yahoo! Research) (Most recent copy) available at:

Randomized Linear Algebra Algorithms
Goal: To develop and analyze (fast) Monte Carlo algorithms for performing useful computations on large matrices and tensors. Matrix Multiplication Computation of the Singular Value Decomposition Computation of the CUR Decomposition Testing Feasibility of Linear Programs Least Squares Approximation Tensor computations: SVD generalizations Tensor computations: CUR generalization Such computations generally require time which is superlinear in the number of nonzero elements of the matrix/tensor, e.g., O(n3) for n £ n matrices.

Randomized Linear Algebra Algorithms
Motivation: (Algorithmic) To speed up computations in applications where extremely large data sets are modeled by matrices/tensors and, e.g., O(n3) time is not an option. (Algorithmic) To reduce the memory requirements in applications where the data sets are modeled by matrices/tensors, and storing the whole data is not an option, because the data sets are extremely large. (Equivalent to the above) To provide some analysis of the performance of the accuracy performance of simple algorithms when only a small sample of the full data set is available. (Structural) To reveal novel structural properties of the datasets, given sufficient computational time.

Example: the CUR decomposition
Carefully chosen U Goal: make (some norm) of A-CUR small. O(1) rows Why? After making two passes over A, we can compute provably good C, U, and R and store them (“sketch”) instead of A: O(m+n) vs. O(n2) space. O(1) columns Why? Given a sample consisting of a few columns (C) and a few rows (R) of A, we can compute U and “reconstruct” A as CUR; if the sampling probabilities are not “too bad”, we get provably good accuracy. Why? Given sufficient time, we can find C, U and R such that A – CUR is “very” small. This might lead to better understanding of the data.

Applications of such algorithms
Matrices arise, e.g., since m objects (documents, genomes, images, web pages), each with n features, may be represented by an m x n matrix A. Covariance Matrices Latent Semantic Indexing DNA Microarray Data Eigenfaces and Image Recognition Similarity Queries Matrix Reconstruction LOTS of other data applications!! More generally, Linear and Nonlinear Programming Applications Design of Approximation Algorithms Statistical Learning Theory Applications

Overview (1/3) Data Streaming Models and Random Sampling
Matrix Multiplication Feasibility testing of Linear Programs Singular Value Decomposition Sampling rows/columns to approximate singular vectors/values Sampling elements to approximate singular vectors/values CUR Matrix Decomposition Lower bounds on sampling complexities

Overview (2/3) Applications of Matrix CUR Tensor-based data sets
Designing approximation algorithms Data mining Microarray data Recommendation Systems Tensor-based data sets Tensor-CUR Applications of Tensor-CUR Hyperspectral data Recommendation systems

Overview (3/3) Kernel-CUR and the Nystrom Method Regression problems
Least squares problems The CUR algorithm revisited A general bound Iterative sampling of rows/columns “Optimal” choices of rows/columns Application: finding ht-SNPs Conclusions & open problems

Computation on Massive Data Sets
Data are too large to fit into main memory; they are either not stored or are stored in external memory. Algorithms that compute on data streams examine the stream, keep a small “sketch” of the data, and perform computations on the sketch. Algorithms are (typically) randomized and approximate. Evaluate performance by measures such as: the time to process an item the number of passes over the data the additional workspace and time the quality of the approximation returned Munro & Paterson ’78: studied “the relation between the amount of internal storage available and the number of passes required to select the k-th highest of n inputs.”

The Pass Efficient Model
Motivation: Amount of disk/tape space has increased enormously; RAM and computing speeds have increased less rapidly. Can store large amounts of data, but Cannot process these data with traditional algorithms. In the Pass-Efficient Model: Data are assumed to be stored on disk/tape. Algorithm has access to the data via a pass over the data. Algorithm is allowed additional RAM space and additional computation time. An algorithm is pass-efficient if it uses a small constant number of passes and sublinear additional time and space to compute a description of the solution. Note: If data are an m x n matrix A, then algorithms which require additional time and space that is O(m+n) or O(1) are pass-efficient.

Random Sampling Random sampling is used to estimate some parameter defined over a very large set by looking at only a very small subset. Uniform Sampling: every piece of data is equally likely to be picked. Advantages: “Coins” can be tossed “blindly.” Even if the number of data elements is not known in advance, can select one element u.a.r. in one pass over the data. Much recent work on quantities that may be approximated with a small uniformly drawn sample. Disadvantages: Many quantities cannot be approximated well with a small random sample that is uniformly drawn. E.g., estimate the sum of N numbers, where one number is very large and the others very small.

Random Sampling (cont’d)
Nonuniform Sampling: Advantages: Can obtain much more generality and big gains, e.g., can approximately solve problems in sparse as well as dense matrices. Smaller sampling complexity for similar bounds. Disadvantages: Must determine the nonuniform probabilities; multiple passes over the data usually needed. Randomized Algorithms for Linear Algebra problems: A “sketch” consisting of a small number of judiciously chosen and randomly sampled rows and columns (or elements) is sufficient for provably rapid and efficient approximation of many matrix operations.

Approximating Matrix Multiplication …
(D. & Kannan FOCS ’01, and D., Kannan, & M. TR ’04, SICOMP ’05) Problem Statement Given an m-by-n matrix A and an n-by-p matrix B, approximate the product A·B, OR, equivalently, Approximate the sum of n rank-one matrices. i-th column of A i-th row of B Each term in the summation is a rank-one matrix

…by random sampling Algorithm
i-th column of A i-th row of B Algorithm Fix a set of probabilities pi, i=1…n, summing up to 1. For t=1 up to s, set jt = i, where Pr(jt = i) = pi; (Pick s terms of the sum, with replacement, with respect to the pi.) Approximate AB by the sum of the s terms, after scaling.

Random sampling (cont’d)
i-th column of A i-th row of B Keeping the terms j1, j2, … js.

The algorithm (matrix notation)
Pick s columns of A to form an m-by-s matrix C and the corresponding s rows of B to form an s-by-p matrix R. (discard A and B) Approximate A · B by C · R.

The algorithm (matrix notation, cont’d)
Create C and R by performing s i.i.d. trials, with replacement. For t=1 up to s, pick a column A(jt) and a row B(jt) with probability Include A(jt)/(spjt)1/2 as a column of C, and B(jt)/(spjt)1/2 as a row of R.

Simple Lemmas The input matrices are given in “sparse unordered representation”; e.g., their non-zero entries are presented as triples (i, j, Aij) in any order. The expectation of CR (element-wise) is AB. Our nonuniform sampling minimizes the variance of the estimator. It is easy to implement the sampling in two passes. If the matrices are dense the algorithm runs in O(smp) time, instead of O(nmp) time, It requires O(sm+sp) RAM space. Does not tamper with the sparsity of the matrices.

Error Bounds For the above algorithm,
If ||AB||F = (||A||F ||B||F), then the above bound is a relative error bound. This happens if there is “not much cancellation” in the multiplication.

Error Bounds (tight concentration)
For the above algorithm, with probability at least 1-, Notice that we removed the expectation (by applying a martingale argument) and having an extra log(1/) factor. (Markov’s inequality would also remove the expectation, introducing an extra 1/ factor.)

Special case: B = AT If B = AT, then the sampling probabilities are
Also, R = CT, and the error bounds are

Special case: B = AT (cont’d)
(Rudelson & Vershynin ’04, Vershynin ’04) Improvement for the spectral norm bound for the special case B = AT. Uses a result of M. Rudelson for random vectors in isotropic position. Tight concentration results can be proven using Talagrand’s theory. The sampling procedure is slightly different; s columns/rows are kept in expectation, i.e., column i is picked with probability

Empirical evaluation: setup
(Data from D. Lewis & E. Cohen, SODA ’97 & J. of Algorithms ’99) Database Dense document-concept matrix A with 5000 documents and 320 concepts. Experiment Our goal was to identify all document-document matches, e.g. compute AAT and identify all entries larger than some threshold  (proximity problem). The algorithm Approximate AAT (using our method) by CCT. Find all entries of CCT that are larger that -,  > 0 (“candidate entries”). Deterministically compute the dot products for “candidate entries”. Return the ones that are larger than  as matches.

Empirical evaluation: results ( = 0.85)

Empirical evaluation: results ( = 0.75)

Experiments: uniform sampling

Random sampling of Linear Programs
(D., Kannan, & M. TR ‘04 & STACS ’05) Let P(i) denote the i th column of the r-by-n matrix P and suppose that the following Linear Program is feasible: Form Q, a random non-uniform subset of {1…n}, with |Q|=q. Q is formed in q i.i.d. trials, where With probability at least 1-, the following LP is feasible as well:

The picture … If feasible … Sample a few variables!
…then feasible (with high probability). Sample a few variables!

Another picture … The i-th constraint is feasible
Relaxing the constraint guarantees feasibility. The bound on db emerges by applying the Matrix Multiplication result on the matrix-vector product Px. Discarding variables “perturbs” the constraint

A converse LP sampling lemma
Let P(i) denote the i th column of the r-by-n matrix P and suppose that the following Linear Program is infeasible: Form Q, a random non-uniform subset of {1…n}, with |Q|=q. Q is formed in q i.i.d. trials, where With probability at least 1-, the following LP is infeasible as well:

The picture … If infeasible … Sample a few variables!
…then infeasible (with high probability). Sample a few variables!

Another picture … The i-th constraint is infeasible
Tightening the constraint guarantees infeasibility. The bound on db emerges by applying the Matrix Multiplication result on the matrix-vector product Px. Discarding variables “perturbs” the constraint

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. S k: diagonal matrix containing the top k singular values of A. Also, Ak=UkUkTA. Ak is a matrix of rank k such that ||A-Ak||2,F is minimized over all rank k matrices! This property of very useful in the context of Principal Component Analysis.

Approximating SVD in O(n) time
(Frieze, Kannan & Vempala FOCS ‘98, D., Frieze, Kannan, Vempala & Vinay SODA ’99, JML ’04, D. Kannan, & M. TR ’04, SICOMP ’05) Given: m x n matrix A Sample c columns from A and rescale to form the m x c matrix C. Compute the m x k matrix Hk of the top k left singular vectors of C. Structural Theorem: For any probabilities and number of columns: ||A-HkHkTA||2,F2 ≤ ||A-Ak||2,F2 + 2√k||AAT-CCT||F Algorithmic Theorem: If pi = |A(i)|2/||A||F2 and c ≥ 42k/2, then: ||A-HkHkTA||2,F2 ≤ ||A-Ak||2,F2 + ||A||F2. Proof: via matrix multiplication theorem and matrix perturbation theory.

Example of randomized SVD
C Title: C:\Petros\Image Processing\baboondet.eps Creator: MATLAB, The Mathworks, Inc. Preview: This EPS picture was not saved with a preview included in it. Comment: This EPS picture will print to a PostScript printer, but not to other types of printers. Original matrix After sampling columns Compute the top k left singular vectors of the matrix C and store them in the 512-by-k matrix Hk.

Example of randomized SVD (cont’d)
Title: C:\Petros\Image Processing\baboondet.eps Creator: MATLAB, The Mathworks, Inc. Preview: This EPS picture was not saved with a preview included in it. Comment: This EPS picture will print to a PostScript printer, but not to other types of printers. A HkHkTA A and HkHkTA are close.

Element-wise sampling
(Achlioptas & McSherry, STOC ’01, JACM ’05) 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). More details: Let pij 2 [0,1] for all i,j. Create the matrix S from A such that: ||A-S||2 is bounded ! (i) the singular values of A and S are close, and (ii, under additional assumptions) the top k left (right) singular vectors of S span a subspace that is close the to subspace spanned by the top k left (right) singular vectors of A.

How to use it Approximating singular values fast:
Zero out (a large number of) elements of A, scale the remaining ones appropriately. Compute the singular values of the resulting sparse matrix using iterative techniques. (Good choice for pij: pij = sAij2/i,j Aij2, where s denotes the expected number of elements that we seek to keep in S.) Note: Each element is kept or discarded independently of the others. Similar ideas have been used to: explain the success of Latent Semantic Indexing (LSI) (Papadimitriou, Raghavan, Tamaki, Vempala, PODS ’98 & Azar, Fiat, Karlin, McSherry, Saia STOC ’01) design recommendation systems (Azar, Fiat, Karlin, McSherry, Saia STOC ’01 ) speedup kernel computations (Achlioptas, McSherry, and Schölkopf, NIPS ’02)

Element-wise vs. row/column sampling
Quite different techniques! Row/column sampling preserves structural properties of the matrices. Element-wise sampling explains how adding noise and/or quantizing the elements of a matrix perturbs its singular values/vectors. The two techniques could be complementary! Some similarities and differences: Similar error bounds. Element-wise sampling is doable in one pass! Running time of element-wise sampling depends on the speed of, e.g., Arnoldi methods.

A novel CUR matrix decomposition
(D. & Kannan, SODA ’03, D., Kannan, & M. TR ’04, SICOMP ’05) Create an approximation to the original matrix of the following form: O(1) columns O(1) rows Carefully chosen U An application: Given a query vector x, instead of computing A · x, compute CUR · x to identify its nearest neighbors,

The CUR decomposition Given a large m-by-n matrix A (stored on disk), compute a decomposition CUR of A such that: C consists of c = O(k/2) columns of A. R consists of r = O(k/2) rows of A. C (R) is created using importance sampling, e.g. columns (rows) are picked in i.i.d. trials with respect to probabilities

The CUR decomposition (cont’d)
Given a large m-by-n matrix A (stored on disk), compute a decomposition CUR of A such that: C, U, R can be stored in O(m+n) space, after making two passes through the entire matrix A, using O(m+n) additional space and time. The product CUR satisfies (with high probability)

Computing U Intuition (which can be formalized):
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 a good fit for the row A(i) using the rows in R as the basis, by approximately solving 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.

Computing U (cont’d) Given c elements of A(i) the algorithm computes a good fit for the row A(i) using the rows in R as the basis, by approximately solving: However, our CUR decomposition approximates the vectors u instead of exactly computing them. Later in this talk: new error bounds using the optimal coefficients.

Error bounds for CUR Assume Ak is the “best” rank k approximation to A (through SVD). Then, if we pick O(k/2) rows and O(k/2) columns, If we pick O(1/2) rows and O(1/2) columns,

Other (randomized) CUR decompositions
Computing U in constant (O(1) instead of O(m+n)) space and time: (D., Kannan, & M. TR ’04, SICOMP ’05) samples O(poly(k,)) rows and columns of A & needs an extra pass. significantly improves the error bounds of Frieze, Kannan, and Vempala, FOCS ’98. Computing U and R for any C: (D., M., & Muthukrishnan ’05) For any subset of the columns, denoted C (e.g., chosen by the practitioner) Obtain bounds of the form: || A - CUR ||F ≤ ( 1 +  ) || A - CC+ A ||F Uses ideas from approximating L2 Regression problems by random sampling. Can combine with recent algorithms/heuristics for choosing columns; more details later.

Other CUR decompositions
(G.W. Stewart, Num. Math. ’99, TR ’04) Compute sparse low-rank approximations to sparse matrices. The quasi-Gram-Schmidt (variant of QR) algorithm: Input: m x n matrix A. Output: m x k matrix C (k columns of A) and upper-triangular k x k matrix SC (that orthogonalizes these columns). Apply this algorithm to A and AT: C = QC SC and RT = QR SR. QC and QR not explicitly computed. Then A ≈ CUR, where U minimizes ||A-CUR||F2 is:

Other CUR decompositions, cont’d
(Goreinov, Tyrtyshnikov, and Zamarashkin, LAA ’97 & Goreinov, Tyrtyshnikov, Cont. Math. ’01) If A is approximately rank-k to accuracy  (for  sufficiently small) then there exist C, R, and W such that Notes: Scattering applications with large matrices with low-rank blocks. Relate to maximum volume concept in interpolation theory. They call the CUR decomposition the pseudoskeleton component of A.

Lower Bounds Question: How many queries does a sampling algorithm need to approximate a given function accurately with high probability? (Ziv Bar-Yossef ’03, ’04) Lower bounds for the low rank matrix approximation problem and the matrix reconstruction problem. Any sampling algorithm that w.h.p. finds a good low rank approximation requires (m+n) queries. Even if the algorithm is given the exact weight distribution over the columns of a matrix it will still require (k/4) queries. Finding a matrix D such that ||A-D||F ≤  ||A||F requires (mn) queries and that finding a D such that ||A-D||2 ≤  ||A||2 requires (m+n) queries. Applied to our results: The LinearTimeSVD algorithm is optimal w.r.t. ||||F bounds. The ConstantTimeSVD algorithm is optimal w.r.t. ||||2 bounds up to poly factors. The CUR algorithm is optimal for constant .

Approximating Max-Cut
Max-Cut (NP-hard for general and dense graphs, Max-SNP) Given a graph G=(V,E), |V|=n, partition V in two disjoint subsets V1 and V2 such that the number of edges of E that have one endpoint in V1 and one endpoint in V2 is maximized. Goemans & Williamson ’94: approximation algorithm (might be tight!) Arora, Karger & Karpinski ’95: LP based, en2 additive error(*) De La Vega ’96: Combinatorial methods, en2 additive error Frieze & Kannan ’96: Linear Algebraic methods, en2 additive error(*) Goldreich, Goldwasser & Ron ’96: Property Testing, en2 additive error Alon, De La Vega, Kannan & Karpinski ’02, ’03 : Cut Decomposition, en2 additive error(*) All the above algorithms run in constant time/space. (*) More generally, it achieves enr for all problems in Max-r-CSP (=Max-SNP)

Weighted Max-Cut via CUR
(D., Kannan, & M. TR ’04 and STACS ’05) Let A denote the adjacency matrix of G. All previous algorithms also guarantee an additive error approximation for the weighted Max-Cut problem, where Amax is the maximum edge weight in the graph. in constant time and space after reading the graph a small number of times. Our result: We can approximate the weighted Max-Cut of G(V,E) up to additive error

The algorithm (sketch)
Let A denote the adjacency matrix of the weighted graph G=(V,E). Then, Step 1: Replace A by the ConstantTimeCUR decomposition. Then, Follows from Compute the MaxCut of CUR, thus getting an approximation to the MaxCut of A. It turns out that we can solve the problem on CUR in constant time, in constant space…

Data mining with CUR 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). E.g., m documents, represented w.r.t. n terms. Queries Given a new object x, find similar objects in the database (nearest neighbors).

Data mining with CUR (cont’d)
Two objects are “close” if the angle between their corresponding (normalized) vectors is small. So, 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.

Data mining with CUR (cont’d)
Assume that CUR is an approximation to A, such that CUR is stored efficiently (e.g. in RAM). Given a query vector x, instead of computing A · x, compute CUR · x to identify its nearest neighbors. The CUR algorithm guarantees a bound on the worst case choice of x. (Also recall how we used the matrix multiplication algorithm to solve the proximity problem, namely find all pairs of objects that are “close”.)

Genetic microarray data and CUR
Exploit structural properties of CUR in biological applications: Experimental conditions Find a “good” set of genes and arrays to include in C and R? Provable and/or heuristic strategies are acceptable. genes Common in Biological/Chemical/Medical applications of PCA: Explain the singular vectors, by mapping them to meaningful biological processes. This is a “challenging” task (think: reification) ! CUR is a low-rank decomposition in terms of the data that practitioners understand. Use it to explain the data and do dimensionality reduction, classification, clustering. Gene microarray data: M., D., & Alter (UT Austin) (sporulation and cell cycle data).

Recommendation Systems
(D., Raghavan, & Kerenidis, STOC ’02) The problem: Assume the existence of m customers and n products. Then, A is an (unknown) matrix s.t. Aij is the utility of product j for customer i. Our goal is to recreate A from a few samples, s.t. we can recommend high utility products to customers. (Kumar, Raghavan, Rajagopalan & Tomkins FOCS ’98) Assuming strong clustering of the products, they offer competitive algorithms even with only 2 samples/customer. (Azar, Fiat, Karlin, McSherry & Saia STOC ’01) Assuming sampling of (mn) entries of A and a certain gap requirement, they -very- accurately recreate A.

Recommendation Systems
The problem: Assume the existence of m customers and n products. Then, A is an (unknown) matrix s.t. Aij is the utility of product j for customer i. Our goal is to recreate A from a few samples, s.t. we can recommend high utility products to customers. Critical assumption (supported by experimental evidence): There exist a small, constant (e.g. k) number of different customer types or, equivalently, the rows of the matrix A are (approximately) linearly dependent or, equivalently, the matrix A has a “good” low-rank (e.g. k) approximation.

Recommendation Systems (cont’d)
Question: Can we get competitive performance by sampling less than (mn) elements? products customers Customer sample (guinea pigs) (purchases, small surveys) Answer: Apply the CUR decomposition.

Details: Sample a constant number of rows and columns of A and compute A’=CUR. Assuming that A has a “good” low rank approximation, This implies that we can make relatively accurate recommendations with far fewer samples.

Datasets modeled as tensors
Goal: Extract structure from a tensor dataset A (naively, a dataset subscripted by multiple indices) using a small number of samples. Mode 2 Mode 1 Mode 3 Q. What do we know about tensor decompositions? A. Not much, although tensors arise in numerous applications. m £ n £ p tensor A

Tensors in Applications
Tensors appear both in Math and CS. Represent high dimensional functions. Connections to complexity theory (i.e., matrix multiplication complexity). Statistical applications (i.e., Independent Component Analysis, higher order statistics, etc.). Large data-set applications (e.g., Medical Imaging & Hyperspectral Imaging) Problem: There does not exist a definition of tensor rank (and associated tensor SVD) with the – nice – properties found in the matrix case. (Lek-Heng Lim ’05: strong impossibility results!) Heuristic solution: “unfold” the tensor along a mode and apply Linear Algebra.

Tensor rank BUT A definition of tensor rank Given a tensor
(Hastad, J of Algorithms ’90, DeLaVega, Kannan, Karpinski, Vempala STOC ’05) A definition of tensor rank Given a tensor find the minimum number of rank one tensors into it can be decomposed. outer product agrees with matrices for d=2 related to computing bilinear forms and algebraic complexity theory some approximations are possible, see DKKV ’05 BUT computing it is NP-hard only weak bounds are known tensor rank depends on the underlying ring of scalars successive rank one approxi-imations are no good

Tensor -rank The -rank of a tensor Given Pros: Easily computable,
Cons: different -ranks are different, information is lost. create the “unfolded” matrix and compute its rank, called the -rank. “unfold”

Tensors in real applications
3 classes of tensors in data applications All modes are comparable (e.g., tensor faces, chemometrics) A priori dominant mode (e.g., coarse scales, microarrays vs. time, images vs. frequency) All other combinations (D. and M. ’05) TensorSVD paper deals with (1). We will focus on (2), where there is a preferred mode.

The TensorCUR algorithm (3-modes)
Choose the preferred mode  (time) Pick a few representative snapshots: Express all the other snapshots in terms of the representative snapshots. n environmental conditions m genes p time/frequency slices 2 samples randomly sample

The TensorCUR algorithm (cont’d)
Let R denote the tensor of the sampled snapshots. Express the remaining images as linear combinations of the sampled snapshots. 2 samples m genes n environmental conditions First, pick a constant number of “fibers” of the tensor A (the red dotted lines). Express the remaining snapshots as linear combination of the sampled snapshots. sampled snapshots sampled fibers p time steps n environmental conditions m genes

The TensorCUR algorithm (cont’d)
Theorem: Unfold R along the a dimension and pre-multiply by CU Best rank ka approximation to A[a] How to proceed: Can recurse on each sub-tensor in R, or do SVD, exact or approximate, or do kernel-based diffusion analysis, or do wavelet-based methods. TensorCUR: Framework for dealing with very large tensor-based data sets, to extract a “sketch” in a principled and optimal manner, which can be coupled with more traditional methods of data analysis.

TensorCUR on scientific data sets
Apply random sampling methodology and kernel-based Laplacian methods to large physical and chemical and biological data sets. Common application areas of large data set analysis: telecommunications, finance, web-based modeling, and astronomy. Scientific data sets are quite different: with respect to their size, with respect to their noise properties, and with respect to the available field-specific intuition.

Data sets being considered
Sequence and mutational data from G-protein coupled receptors to identify mutants with enhanced stability properties. Genomic microarray data to understand large-scale genomic and cellular behavior. Hyperspectral colon cancer data for improved detection of anomalous behavior. Time-resolved fMRI data to better represent large, complex visual brain-imaging data. Simulational data to more efficiently conduct large scale computations.

Sampling hyperspectral data
Sample slabs depending on total absorption: Sample fibers uniformly (since intensity depends on stain).

Look at the exact 65-th slab.

The 65-th slab approximately reconstructed
This slab was reconstructed by approximate least-squares fit to the basis from slabs 41 and 50, using 1000 (of 250K) pixels/fibers.

Tissue Classification - Exact Data

Tissue Classification - Ns=12 & Nf=1000

TensorCUR on Recommendation Systems
Our previous setup: Assume the existence of m customers and n products. Then, A is an (unknown) matrix s.t. Aij is the utility of product j for customer i. Our goal is to recreate A from a few samples, s.t. we can recommend high utility products to customers. products Customer sample (guinea pigs) customers Customer sample (purchases, small surveys)

Comment: It is folklore knowledge in economics literature that product utility is an ordinal and not a cardinal quantity. Thus, it is more natural to compare products than assign utility values. Model revisited: Every customer has an n-by-n matrix (whose entries are §1) and represent pairwise product comparisons. Overall, there are m such matrices, forming an n-by-n-by-m 3-mode array or a three-dimensional tensor, denoted by A. We seek to extract the “structure” of this tensor by sampling.

Our goal: Recreate the tensor A from a few samples in order to recommend high utility products to the customers. n products m customers (Similar to previous discussion.)

Motivation for Kernels (1 of 3)
Methods to extract linear structure from the data: Support Vector Machines (SVMs). Gaussian Processes (GPs). Singular Value Decomposition (SVD) and the related PCA. Kernel-based learning methods to extract non-linear structure: Choose features to define a (dot product) space F. Map the data, X, to F by : X ! F. Do classification, regression, and clustering in F with linear methods.

Use dot products for information about mutual positions. Define the kernel or Gram matrix: Gij=kij=((X(i)), (X(j))). Algorithms that are expressed in terms of dot products may be given as input the Gram matrix G instead of the data covariance matrix XTX. Note: Isomap, LLE, graph Laplacian eigenmaps, Hessian eigenmaps, SDE (dimensionality reduction methods for nonlinear manifolds) are kernel-PCA for particular Gram matrices. Note: for Mercer kernels, G is SPSD.

If the Gram matrix G, where Gij=kij=((X(i)), (X(j))), is dense but has low numerical rank, then calculations of interest still need O(n2) space and O(n3) time: matrix inversion in GP prediction, quadratic programming problems in SVMs, computation of eigendecomposition of G. Relevant recent work using low-rank methods: (Achlioptas, McSherry, and Schölkopf, NIPS ’02), randomized kernels. Nystrom method for out-of-sample extensions (Williams and Seeger, NIPS ’01 and others),.

Kernel-CUR Main algorithm:
(D. & M., COLT ’05, TR ’05) Main algorithm: Randomized algorithm to approximate a Gram matrix. Low-rank approximation in terms of columns (and rows) of G=XTX. Main quality-of-approximation theorem: Provably good approximation if nonuniform probabilities are used. Discussion of the Nystrom method: Nystrom method for integral equations and matrix problems. Relationship to randomized SVD and CUR algorithms.

Kernel-CUR Algorithm Input: n x n SPSD matrix G, probabilities {pi, 1=1,…,n}, c <= n, and k <= c. Output: n x c matrix C, and c x c matrix Wk+ (s.t. CWk+CT ≈ G). Algorithm: Pick c columns of G in i.i.d. trials, with replacement and with respect to the {pi}; let L be the set of indices of the sampled columns. Scale each sampled column (with index i 2 L) by dividing its by (cpi)1/2. Let C be the n x c matrix containing the rescaled sampled columns. Let W be the c x c submatrix of G with entries Gij/(cpi1/2pj1/2), i,j 2 L. Compute Wk+.

Notes on the algorithm Note the structural simplicity of the algorithm: C consists of a small number of representative data points. W consists of the induced subgraph defined by those points. Computational resource requirements: Assume the data X (or Gram matrix G) are stored externally. Algorithm performs two passes over the data. Algorithm uses O(n) additional scratch space and additional computation time.

Analysis of Kernel-CUR
Let  > 0 and  = O(log(1/)). Construct an approximation CWk+CT with the Kernel-CUR algorithm by sampling c columns of G with probabilities pi = Gii2/ i Gii2. If c = O(k log(1/)/4), then with probability at least 1-, If c = O(log(1/)/4), then with probability at least 1-,

Interpreting the sampling probabilities
If the sampling probabilities were: pi = |G(i)|2/||G||F2 they would provide a bias towards data points that are more “important” - longer and/or more representative. the additional error would be ||G||F and not  i Gii2= ||X||F2 (where G=XTX). Our sampling probabilities ignore correlations: pi = Gii2/ i Gii2 = |X(i)|2/||X||F2

The Nystrom Method (1 of 3)
Consider the eigenfunction problem: Discretize with a quadrature rule ({wj} are the weights and {sj} are the quadrature points): Choose a set {xi} of Nystrom points (often the same as {si}): Extend the eigenvectors on the Nystrom points on D:

Randomized SVD Algorithms Randomly sample columns (or rows). Compute/approximate low-dimensional singular vectors. Nystrom-extend to approximate Hk, the high-dimensional singular vectors. Bound ||A-HkHkTA||2,F <= ||A-Ak||2,F + ||A||F. Randomized CUR Algorithms Randomly sample columns and rows. Bound ||A-CUR||2,F <= ||A-Ak||2,F + ||A||F. Do not need or use the SPSD property.

Fast Computation with Kernels
Adjacency matrix, t=0 To construct landmarks, randomly sample with the “right” probabilities: for outliers, uniform sampling. Adjacency matrix, t=t* Kernel-based diffusion To construct a coarse-grained version of the data graph: Construct landmarks, Partition/Quantization, Diffusion wavelets.

Regression problems (D., M. & Muthukrishnan, ’05) We seek sampling-based algorithms for solving l2 regression. We are interested in overconstrained problems, n >> d. Typically, there is no x such that Ax = b.

“Induced” regression problems
sampled “rows” of b sampled rows of A scaling to account for undersampling

Regression problems, definition
There is work by K. Clarkson in SODA 2005 on sampling-based algorithms for l1 regression ( = 1 ) for overconstrained problems.

Projection of b on the subspace spanned by the columns of A
Exact solution Projection of b on the subspace spanned by the columns of A Pseudoinverse of A

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. : rank of A. Computing the SVD takes O(d2n) time. The pseudoinverse of A is

Can sampling methods provide accurate estimates for l2 regression?
Questions … Can sampling methods provide accurate estimates for l2 regression? Is it possible to approximate the optimal vector and the optimal value value Z2 by only looking at a small sample from the input? (Even if it takes some sophisticated oracles to actually perform the sampling …) Equivalently, is there an induced subproblem of the full regression problem, whose optimal solution and its value Z2,s aproximates the optimal solution and its value Z2?

Creating an induced subproblem
Algorithm Fix a set of probabilities pi, i=1…n, summing up to 1. Pick r indices from {1…n} in r i.i.d. trials, with respect to the pi’s. For each sampled index j, keep the j-th row of A and the j-th element of b; rescale both by (1/rpj)1/2.

The induced subproblem
sampled rows of A, rescaled sampled elements of b, rescaled

Our results If the pi satisfy certain conditions, then with probability at least 1-, The sampling complexity is

Our results, cont’d If the pi satisfy certain conditions, then with probability at least 1-, (A): condition number of A The sampling complexity is

Back to induced subproblems …
sampled elements of b, rescaled sampled rows of A, rescaled The relevant information for l2 regression if n >> d is contained in an induced subproblem of size O(d2)-by-d. (upcoming writeup: we can reduce the sampling complexity to r = O(d).)

Conditions on the probabilities, SVD
U (V): orthogonal matrix containing the left (right) singular vectors of A. S: diagonal matrix containing the singular values of A. : rank of A. Let U(i) denote the i-th row of U. Let U? 2 Rn £ (n -) denote the orthogonal complement of U.

Conditions for the probabilities
The conditions that the pi must satisfy, for some 1, 2, 3 2 (0,1]: lengths of rows of matrix of left singular vectors of A Component of b not in the span of the columns of A The sampling complexity is: Small i ) more sampling

Computing “good” probabilities
In O(nd2) time we can easily compute pi’s that satisfy all three conditions, with 1 = 2 = 3 = 1/3. (Too expensive!) Open question: can we compute “good” probabilities faster, in a pass efficient manner? Some assumptions might be acceptable (e.g., bounded condition number of A, etc.)

Revisiting the CUR decomposition
(D., M. & Muthukrishnan, ’05) Carefully chosen U Create an approximation to A, using rows and columns of A O(1) rows O(1) columns Goal: provide (good) bounds for some norm of the error matrix A – CUR How do we draw the rows and columns of A to include in C and R? How do we construct U?

A simple algorithm Carefully chosen U Create an approximation to A, using rows and columns of A O(1) rows O(1) columns Algorithm Step 1: pick a few columns of A and include them in C (“basis” columns). (Any set of columns will do; “good” choices will be discussed later) Step 2: express all columns of A as linear combinations of the “basis” columns.

Step 2 “basis” columns i-th column The second step is trivial; it amounts to solving an l2 regression problem. Unfortunately, it necessitates looking at every element of A(i) and expresses A as CC+A. Can we approximately find the vector of coefficients by looking only at a few elements of A(i)? Yes!

D: rescaling, diagonal matrix
Step 3 i-th column of R D: rescaling, diagonal matrix sampled rows of C Pick a few ( r ) rows of A and solve “induced” sampled l2 regression problems using these rows for all i 2 {1…n} (every column of A). (Not straightforward from the l2 regression results.)

Error bounds (given C) In O(c2m + cmn) = O(mn) time, we can construct D and R such that holds with probability at least 1-. We need to pick r= O(c2 log(1/)/2) rows. (upcoming writeup: we can reduce c2 to c.) Constructing D and R: we compute a set of probabilities pi and sample and rescale rows of A in r i.i.d. trials with respect to the pi’s.

Forming the pi (1/3) U (V): orthogonal matrix containing the left (right) singular vectors of C. S: diagonal matrix containing the singular values of C. : rank of C. Let U(i) denote the i-th row of U. Let U? 2 Rm £ (m -) denote the orthogonal complement of U.

Small i ) more sampling
Forming the pi (2/3) Compute pi that satisfy, for some 1, 2, 3 2 (0,1]: lengths of rows of matrix of left singular vectors of C Component of A not in the span of the columns of C The sampling complexity is: Small i ) more sampling

Forming the pi (3/3) In O(c2m + mnc) = O(mn) time we can easily compute pi that satisfy all the constraints, with 1, 2, 3 all equal to 1/3.

Overall decomposition
columns of A diagonal rescaling matrix (r £ c) rows of A ”intersection” of C and R (r £ c)

Overall decomposition (a variant)
columns of A rows of A ”intersection” of C and R (Goreinov, Tyrtyrshnikov, & Zamarashkin LAA ’98) Error bounds if C and R contain the columns and rows of A that define a parallelpiped of maximal volume.

A (first) choice for C (D., Frieze, Kannan, Vempala & Vinay ’99, ‘03 and D., Kannan, & M. TR ‘04 & SICOMP ‘05) (For any k) Assume Ak is the “best” rank k approximation to A (through SVD). Then, after making two passes through A, we can construct a matrix C such that, with probability at least 1-, We need to pick O( k log(1/) / 2 ) columns to include in C. (Columns are picked with probability proportional to their Euclidean length squared; recall the SVD algorithm early in this tutorial.)

O( k log(1/) / 2 ) columns, and
CUR error (For any k) In O(mn) time, we can construct C, U, and R such that holds with probability at least 1-, by picking O( k log(1/) / 2 ) columns, and O( k2 log3(1/) / 6 ) rows.

A (second) choice for C (Rademacher, Vempala & Wang ’04, D. & M. TR ’05) (For any k) Assume Ak is the “best” rank k approximation to A (through SVD). Then, after making 2t passes through A, we can construct a matrix C such that, with probability at least 1-, We need to pick columns of A in t rounds, picking O(k log(1/) / 2) columns per round.

The MultiPass sampling algorithm
R = A; S = empty set; For i=1…t Pick c columns from A0; each column is sampled with probability proportional to its Euclidean length squared. S = S [ {sampled columns from step 1}; R = A – PS A; End For In words, we subtract the projection of A on the subspace spanned by the columns that are included in the sample.

Intuition R = A; S = empty set; For i=1…t
Pick c columns from A0; each column is sampled with probability proportional to its Euclidean length squared. S = S [ {sampled columns from step 1}; R = A – PS A; End For A simple idea: this step allows us to focus on the subspace of A that is not “sufficiently” spanned by the current selection of columns.

O( t k log(1/) / 2 ) columns, and
CUR error (For any k) In O(mnt) time, we can construct C, U, and R such that holds with probability at least 1-, by picking O( t k log(1/) / 2 ) columns, and O( t2 k2 log3(1/) / 6 ) rows.

A (third) choice for C (Rademacher, Vempala & Wang ’04) (For any k) Assume Ak is the “best” rank k approximation to A (through SVD). Then, there exist c = O(k2/2) columns of A such that ( It seems possible to use the volume work by Goreinov et. al. to design randomized strategies to pick such columns. )

CUR error (For any k) In O(nc + mn) time, we can construct C, U, and R such that holds with probability at least 1-, by picking O( k2 / 2 ) columns, and O( k4 log(1/) / 6 ) rows. Notice that we can pick the rows efficiently, given the columns.

Example: finding ht-SNPs
(data from K. Kidd’s lab at Yale University, joint work with Dr. Paschou at Yale University) Single Nucleotide Polymorphisms: the most common type of genetic variation in the genome. (Locations at the human genome where typically two alternate nucleotide bases are observed.) Why are SNPs important: they occur quite frequently within the genome allowing the tracking of disease genes and population histories. Thus, they are effective markers for genomic research! There are ∼10 million SNPs in the human genome (under some assumptions).

Research topics Research Topics (working within a specific chromosomal region of the human genome): (i) Are different SNPs correlated, either within a population, or across different populations? (Folklore knowledge: yes). (ii) Find a “good” set of haplotype tagging-SNPs capturing the diversity of a chromosomal region of the human genome. (iii) Is extrapolation feasible? (Recall Nystrom/CUR extrapolation.) This would save (a lot of) time and money! Existing literature Pairwise metrics of SNP correlation, called LD (linkage disequilibrium) distance, based on nucleotide frequencies and co-occurrences. Almost no metrics exist for measuring correlation between more than 2 SNPs and LD is very difficult to generalize. Exhaustive and semi-exhaustive algorithms in order to pick “good” ht-SNPs that have small LD distance with all other SNPs. Using Linear Algebra: an SVD based algorithm was proposed by Lin & Altman, Am. J. Hum. Gen

The raw data Notes: Each SNP genotype consists of two alleles.
SNPs AG CT GT GG CT CC CC CC CC AG AG AG AG AG AA CT AA GG GG CC GG AG CG AC CC AA CC AA GG TT AG CT CG CG CG AT CT CT AG CT AG GG GT GA AG GG TT TT GG TT CC CC CC CC GG AA AG AG AG AA CT AA GG GG CC GG AA GG AA CC AA CC AA GG TT AA TT GG GG GG TT TT CC GG TT GG GG TT GG AA GG TT TT GG TT CC CC CC CC GG AA AG AG AA AG CT AA GG GG CC AG AG CG AC CC AA CC AA GG TT AG CT CG CG CG AT CT CT AG CT AG GG GT GA AG GG TT TT GG TT CC CC CC CC GG AA AG AG AG AA CC GG AA CC CC AG GG CC AC CC AA CG AA GG TT AG CT CG CG CG AT CT CT AG CT AG GT GT GA AG GG TT TT GG TT CC CC CC CC GG AA GG GG GG AA CT AA GG GG CT GG AA CC AC CG AA CC AA GG TT GG CC CG CG CG AT CT CT AG CT AG GG TT GG AA GG TT TT GG TT CC CC CG CC AG AG AG AG AG AA CT AA GG GG CT GG AG CC CC CG AA CC AA GT TT AG CT CG CG CG AT CT CT AG CT AG GG TT GG AA GG TT TT GG TT CC CC CC CC GG AA AG AG AG AA TT AA GG GG CC AG AG CG AA CC AA CG AA GG TT AA TT GG GG GG TT TT CC GG TT GG GT TT GG AA individuals Notes: Each SNP genotype consists of two alleles. Observed SNPs are in known positions in the human genome. Exactly two nucleotides appear in each column.

Encoding the data SNPs individuals How? Exactly two nucleotides (out of A,G,C,T) appear in each column. Thus, the two alleles might be both equal to the first one (encode by +1), both equal to the second one (encode by -1), or different (encode by 0). Notes Order of the alleles is irrelevant, so TG is the same as GT. Encoding, e.g., GG to +1 and TT to -1 is not any different (for our purposes) from encoding GG to -1 and TT to +1. (Recall that flipping the signs of the columns of a matrix only changes the signs of the left singular vectors.)

Evaluating (linear) structure
For each population We examined samples from 38 different populations, with an average size of 50 subjects. For each subject, we were given 63 SNPs, covering a chromosomal region of the human genome which is ¼ 900,000 bases long, close to the end of the long arm of chromosome 17. For each population, we ran SVD to determine the “optimal” number k of principal components that are necessary in order to cover 90% of its spectrum. Expressing columns of A as linear combinations of the top k left singular vectors This means that if we select the top k left singular vectors Uk we can express every column (i.e, SNP) of A as a linear combination of the left singular vectors loosing at most 10% of the information in the matrix: BUT: Uk consists of “superpositions” of the columns of A. Is it possible to pick a small number (e.g., roughly k) of columns of A and express every column (i.e., SNP) of A as a linear combination of the picked columns (matrix C) loosing at most 10% of the information in the matrix? Expressing columns of A as linear combinations of a few columns of A

Algorithms We ran various schemes to select “good” columns (i.e., ht-SNPs). A greedy Multi-Pass heuristic scheme gave the best results. (Select one column in each round, subtracting from A the projection of A on this column and repeating.) Notice that some provable results exist for similar algorithms. Nice feature: SVD provides a non-trivial (maybe not achievable) lower bound. In many cases, the lower bound is attained by the greedy heuristic! (In our data, at most k+5 columns suffice to extract 90% of the structure.)

America Oceania Asia Europe Africa

Extrapolation Given a small number of SNPs for all subjects, and all SNPs for some subjects, extrapolate the values of the missing SNPs. SNPs “Training” data (for a few subjects, we are given all SNPs) individuals SNP sample (for all subjects, we are given a small number of SNPs)

Extrapolation We split our data in “training ” (90%) and “test” (10%).
The training set corresponds to R; for the “test” set we are given data on a small number of SNPs, picked using our greedy multipass heuristic on the “training” set. SNPs individuals “Training” data (for a few subjects, we are given all SNPs) SNP sample (for all subjects, we are given a small number of SNPs)

Rondonian Surui

Arizona Pima

Atayal

Samaritan

European American

African American

Conclusions & Open Problems
Empirical evaluation of different sampling probabilities Uniform Row/column lengths squared Lengths of the rows of the left/right singular vectors Inverse squares of row/column lengths to pick outliers Others, depending on the problem to be solved? Empirical evaluation for different learning problems Matrix reconstruction, regression, classification, clustering, etc. Use CUR for improved interpretability for data matrices Structural vs. algorithmic issues Use CW+CT for improved interpretability of Gram Matrices

Conclusions & Open Problems
Impose other structural properties in CUR-type decompositions Non-negativity Element quantization, e.g., to 0,1,-1 Block-SVD type structure Robust heuristics and robust extensions Especially for noisy data Quasi Monte-Carlo vs. Monte-Carlo vs. Deterministic heuristics of the sketch for different data sets Especially for not-so-large datasets Relate the pass-efficient framework to practice passes over the data a priori knowledge regarding data generation oracle based models

TUTORIAL: Randomized Algorithms for Matrices and Massive Data Sets

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TUTORIAL: Randomized Algorithms for Matrices and Massive Data Sets

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