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TUTORIAL Randomized Algorithms for Matrices and

Massive Data Sets

Michael W. Mahoney Yale University Dept. of

Mathematics (now at Yahoo! Research)

Petros Drineas RPI Dept. of Computer Science

(Most recent copy) available at

http//www.cs.yale.edu/homes/mmahoney http//www.

cs.rpi.edu/drinep

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

Goal make (some norm) of A-CUR small.

Why? After making two passes over A, we can

compute provably good C, U, and R and store them

(sketch) instead of A O(mn) vs. O(n2) space.

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
- 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(mn) 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 (contd)

- 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.

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

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 AB, OR,

equivalently, Approximate the sum of n rank-one

matrices.

Each term in the summation is a rank-one matrix

by random sampling

- Algorithm
- Fix a set of probabilities pi, i1n, summing up

to 1. - For t1 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 (contd)

Keeping the terms j1, j2, js.

The algorithm (matrix notation)

The algorithm (matrix notation, contd)

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(smsp) RAM space.
- Does not tamper with the sparsity of the matrices.

Error Bounds

For the above algorithm,

If ABF ?(AF BF), 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. (Markovs 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 (contd)

(Rudelson Vershynin 04, Vershynin 04)

Improvement for the spectral norm bound for the

special case B AT.

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

?-?, ? gt 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

The picture

Sample a few variables!

Another picture

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

The picture

Sample a few variables!

Another picture

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

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(minmn2 ,

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, AkUkUkTA.

Ak is a matrix of rank k such that A-Ak2,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-HkHkTA2,F2

A-Ak2,F2 2vkAAT-CCTF Algorithmic

Theorem If pi A(i)2/AF2 and c 4?2k/?2,

then A-HkHkTA2,F2 A-Ak2,F2

?AF2. Proof via matrix multiplication

theorem and matrix perturbation theory.

Example of randomized SVD

A

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 (contd)

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).

A-S2 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.

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

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

The CUR decomposition

Given a large m-by-n matrix A (stored on disk),

compute a decomposition CUR of A such that

The CUR decomposition (contd)

- 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(mn) space, after

making two passes through the entire matrix A,

using O(mn) 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 (contd)

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,

Other (randomized) CUR decompositions

- Computing U in constant (O(1) instead of O(mn))

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).

Other CUR decompositions, contd

(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 ?(mn) 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-DF ? AF

requires ?(mn) queries and that finding a D such

that A-D2 ? A2 requires ?(mn) 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 ?.

Overview (2/3)

- Applications of Matrix CUR
- Designing approximation algorithms
- Data mining
- Microarray data
- Recommendation Systems
- Tensor-based data sets
- Tensor-CUR
- Applications of Tensor-CUR
- Hyperspectral data
- Recommendation systems

Approximating Max-Cut

Max-Cut (NP-hard for general and dense graphs,

Max-SNP) Given a graph G(V,E), Vn, 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 .878-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.

The algorithm (sketch)

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 (gt106) objects and n(gt105)

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 (contd)

Two objects are close if the angle between

their corresponding (normalized) vectors is

small. So, xTd cos(x,d) is high when the two

objects are close. Ax 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 gt

Threshold.

Data mining with CUR (contd)

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.

(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 (contd)

Question Can we get competitive performance by

sampling less than ?(mn) elements?

Recommendation Systems (contd)

Details Sample a constant number of rows and

columns of A and compute ACUR. Assuming that A

has a good low rank approximation, This

implies that we can make relatively accurate

recommendations with far fewer samples.

Overview (2/3)

- Applications of Matrix CUR
- Designing approximation algorithms
- Data mining
- Microarray data
- Recommendation Systems
- Tensor-based data sets
- Tensor-CUR
- Applications of Tensor-CUR
- Hyperspectral data
- Recommendation systems

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.

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

(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.

- 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.

The TensorCUR algorithm (contd)

2 samples

- Let R denote the tensor of the sampled

snapshots. - Express the remaining images as linear

combinations of the sampled snapshots.

m genes

n environmental conditions

The TensorCUR algorithm (contd)

Theorem

Unfold R along the a dimension and pre-multiply

by CU

Best rank ka approximation to Aa

- 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.

- 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.

Overview (2/3)

- Applications of Matrix CUR
- Designing approximation algorithms
- Data mining
- Microarray data
- Recommendation Systems
- Tensor-based data sets
- Tensor-CUR
- Applications of Tensor-CUR
- Hyperspectral data
- Recommendation systems

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.

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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 - Ns12 Nf1000

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)

Recommendation Systems (contd)

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.

Recommendation Systems (contd)

Our goal Recreate the tensor A from a few

samples in order to recommend high utility

products to the customers.

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

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.

Motivation for Kernels (2 of 3)

- Use dot products for information about mutual

positions. - Define the kernel or Gram matrix

Gijkij(?(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.

Motivation for Kernels (3 of 3)

- If the Gram matrix G, where Gijkij(?(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

(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 GXTX. - 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,

11,,n, c lt n, and k lt c. Output n x c

matrix C, and c x c matrix Wk (s.t. CWkCT 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 ? gt 0 and ? O(log(1/?)). Construct an

approximation CWkCT 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-?,

Interpreting the sampling probabilities

- If the sampling probabilities were
- pi G(i)2/GF2
- they would provide a bias towards data points

that are more important - longer and/or more

representative. - the additional error would be ?GF and not ?

?i Gii2 ?XF2 (where GXTX).

Our sampling probabilities ignore correlations

pi Gii2/ ?i Gii2 X(i)2/XF2

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)

The Nystrom Method (2 of 3)

The Nystrom Method (3 of 3)

- 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-HkHkTA2,F lt A-Ak2,F

?AF. - Randomized CUR Algorithms
- Randomly sample columns and rows.
- Bound A-CUR2,F lt A-Ak2,F ?AF.
- Do not need or use the SPSD property.

Fast Computation with Kernels

Adjacency matrix, t0

Kernel-based diffusion

- To construct a coarse-grained version of the data

graph - Construct landmarks,
- Partition/Quantization,
- Diffusion wavelets.

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

Regression problems

(D., M. Muthukrishnan, 05)

We seek sampling-based algorithms for solving l2

regression. We are interested in overconstrained

problems, n gtgt d. Typically, there is no x such

that Ax b.

Induced regression problems

sampled rows of b

sampled rows of A

Regression problems, definition

There is work by K. Clarkson in SODA 2005 on

sampling-based algorithms for l1 regression (?

1 ) for overconstrained problems.

Exact solution

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.

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, i1n, summing up

to 1. - Pick r indices from 1n in r i.i.d. trials,

with respect to the pis. - 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

Our results

If the pi satisfy certain conditions, then with

probability at least 1-?,

Our results, contd

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

gtgt 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

Computing good probabilities

In O(nd2) time we can easily compute pis 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.)

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

Revisiting the CUR decomposition

(D., M. Muthukrishnan, 05)

Create an approximation to A, using rows and

columns of A

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

Create an approximation to A, using rows and

columns of A

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 CCA.

Can we approximately find the vector of

coefficients by looking only at a few elements of

A(i)? Yes!

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 1n (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 pis.

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.

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

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

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.)

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 i1t
- 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

Intuition

- R A S empty set
- For i1t
- 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.

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. Kidds 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. 2004.

The raw data

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

0 0 0 1 0 -1 1 1 1 0 0 0 0 0 1 0

1 -1 -1 1 -1 0 0 0 1 1 1 1 -1 -1 0 0 0

0 0 0 0 0 0 0 0 1 0 0 0 -1 -1 -1 1

-1 -1 1 1 1 -1 1 0 0 0 1 0 1 -1 -1 1

-1 1 -1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 -1

-1 1 -1 -1 -1 1 -1 -1 1 -1 -1 -1 1 -1 -1 1

1 1 -1 1 0 0 1 0 0 1 -1 -1 1 0 0 0 0

1 1 1 1 -1 -1 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 -1 -1 -1 1 -1 -1 1 1 1 -1 1

0 0 0 1 1 -1 1 1 1 0 -1 1 0 1 1 0 1

-1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 -1 -1 -1 1 -1 -1 1 1 1 -1 1 -1 -1 -1 1

0 1 -1 -1 0 -1 1 1 0 0 1 1 1 -1 -1 -1 1

0 0 0 0 0 0 0 0 0 1 -1 -1 1 -1 -1 -1

1 -1 -1 1 0 1 0 0 0 0 0 1 0 1 -1 -1 0

-1 0 1 -1 0 1 1 1 -1 -1 0 0 0 0 0 0

0 0 0 0 0 1 -1 -1 1 -1 -1 -1 1 -1 -1 1

1 1 -1 1 0 0 0 1 -1 1 -1 -1 1 0 0 0 1

1 1 0 1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 -1 -1

-1 0 -1 -1 1

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.

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 k5 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.

Rondonian Surui

Arizona Pima

Atayal

Samaritan

European American

African American

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

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 CWCT for improved interpretability of Gram

Matrices - Structural vs. algorithmic issues

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