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Subspace Embeddings for the L1 norm with Applications Christian Sohler TU Dortmund David Woodruff IBM Almaden Subspace Embeddings for the L1 norm with Applications to... Robust Regression and Hyperplane Fitting Outline Massive data sets Regression analysis Our results Our techniques Concluding remarks 3 Massive data sets Examples Internet traffic logs Financial data etc. Algorithms Want nearly linear time or less Usually at the cost of a randomized approximation 4 Regression analysis Regression Statistical method to study dependencies between variables in the presence of noise. 5 Regression analysis Linear Regression Statistical method to study linear dependencies between variables in the presence of noise. 6 Regression analysis Linear Regression Statistical method to study linear dependencies between variables in the presence of noise. Example Ohm's law V = R ∙ I Example Regression 250 200 150 Example Regression 100 50 0 0 50 100 150 7 Regression analysis Linear Regression Statistical method to study linear dependencies between variables in the presence of noise. Example Regression Example 250 Ohm's law V = R ∙ I Find linear function that best 200 fits the data 150 Example Regression 100 50 0 0 50 100 150 8 Regression analysis Linear Regression Statistical method to study linear dependencies between variables in the presence of noise. Standard Setting One measured variable b A set of predictor variables a 1,…, ad Assumption: b = x 0+ a 1 x 1+ … + a d x d + e e is assumed to be noise and the xi are model parameters we want to learn Can assume x0 = 0 Now consider n measured variables 9 Regression analysis Matrix form Input: nd-matrix A and a vector b=(b1,…, bn) n is the number of observations; d is the number of predictor variables Output: x* so that Ax* and b are close Consider the over-constrained case, when n À d Can assume that A has full column rank 10 Regression analysis Least Squares Method Find x* that minimizes S (bi – <Ai*, x*>)² Ai* is i-th row of A Certain desirable statistical properties Method of least absolute deviation (l1 -regression) Find x* that minimizes S |bi – <Ai*, x*>| Cost is less sensitive to outliers than least squares 11 Regression analysis Geometry of regression We want to find an x that minimizes |Ax-b|1 The product Ax can be written as A*1x1 + A*2x2 + ... + A*dxd where A*i is the i-th column of A This is a linear d-dimensional subspace The problem is equivalent to computing the point of the column space of A nearest to b in l1-norm 12 Regression analysis Solving l1 -regression via linear programming Minimize (1,…,1) ∙ (a+ + a ) Subject to: A x + a+- a- = b a+, a- ≥ 0 Generic linear programming gives poly(nd) time Best known algorithm is nd5 log n + poly(d/ε) [Clarkson] 13 Our Results A (1+ε)-approximation algorithm for l1-regression problem Time complexity is nd1.376 + poly(d/ε) (Clarkson’s is nd5 log n + poly(d/ε)) First 1-pass streaming algorithm with small space (poly(d log n /ε) bits) Similar results for hyperplane fitting 14 Outline Massive data sets Regression analysis Our results Our techniques Concluding remarks 15 Our Techniques Notice that for any d x d change of basis matrix U, minx in Rd |Ax-b|1 = minx in Rd |AUx-b|1 16 Our Techniques Notice that for any y 2 Rd, minx in Rd |Ax-b|1 = minx in Rd |Ax-b+Ay|1 We call b-Ay the “residual”, denoted b’, and so minx in Rd |Ax-b|1 = minx in Rd |Ax-b’|1 17 Rough idea behind algorithm of Clarkson Takes nd5 log n time Takes nd5 log n time minx in Rd |Ax-b|1 = minx in Rd |AUx – b’|1 Compute well-conditioned Compute poly(d)Sample poly(d/ε) rows of AU◦b’basis approximation proportional to their l1-norm. Find a basis U so that for all x in Find y such that from the Rd, |Ay-b|1 · poly(d) Sample minx in Rdrows |Ax-b| 1 and Let b’ = b-Aywell-conditioned be the residual|x|basis 1/poly(d) · |AUx|1 · poly(d) |x|1 the residual of the poly(d)approximation Now generic linear Takes poly(d/ε) time Takes nd time programming is efficient Solve l1-regression on the sample, obtaining vector x, and output x 18 Our Techniques Suffices to show how to quickly compute 1. A poly(d)-approximation 2. A well-conditioned basis 19 Our main theorem Theorem There is a probability space over (d log d) n matrices R such that for any nd matrix A, with probability at least 99/100 we have for all x: |Ax|1 ≤ |RAx|1 ≤ d log d ∙ |Ax|1 Embedding is linear is independent of A preserves lengths of an infinite number of vectors 20 Application of our main theorem Computing a poly(d)-approximation Compute RA and Rb Solve x’ = argminx |RAx-Rb|1 Main theorem applied to A◦b implies x’ is a d log d – approximation RA, Rb have d log d rows, so can solve l1-regression efficiently Time is dominated by computing RA, a single matrix-matrix product 21 Application of our main theorem Computing a well-conditioned basis 1. Compute RA Life is really simple! 2. Compute U so that RAU is orthonormal (in the l2-sense) 3. Output AU Time dominated AU is well-conditioned because: by computing RA and AU, two matrix-matrix products |AUx|1 · |RAUx|1 · (d log d)1/2 |RAUx|2 = (d log d)1/2 |x|2 · (d log d)1/2 |x|1 and |AUx|1 ¸ |RAUx|1/(d log d) ¸ |RAUx|2/(d log d) = |x|2/(d log d) ¸ |x|1 /(d3/2 log d) 22 Application of our main theorem It follows that we get an nd1.376 + poly(d/ε) time algorithm for (1+ε)-approximate l1-regression 23 What’s left? We should prove our main theorem Theorem: There is a probability space over (d log d) n matrices R such that for any nd matrix A, with probability at least 99/100 we have for all x: |Ax|1 ≤ |RAx|1 ≤ d log d ∙ |Ax|1 R is simple The entries of R are i.i.d. Cauchy random variables 24 Cauchy random variables pdf(z) = 1/(π(1+z)2) for z in (-1, 1) Infinite expectation and variance z 1-stable: If z1, z2, …, zn are i.i.d. Cauchy, then for a 2 Rn, a1¢z1 + a2¢z2 + … + an¢zn » |a|1¢z, where z is Cauchy 25 main Proof i |Zi| of = (d logtheorem d) with probability 1-exp(-d) by Chernoff By 1-stability, For all rowsonr of R,| |Ax|1 = 1} ε-net argument {Ax shows |RAx| = |Ax| ¢(d log d) for all <r,1 Ax> » 1|Ax| 1¢Z, x where Z is a Cauchy z Scale 1/(d ¢log d) RAxR»by (|Ax| 1 Z1, …, |Ax|1 ¢ Zd log d), where Z1, …, Zd log d are i.i.d. Cauchy |RAx|1 = |Ax|1 i |Zi| / (d log d) But i |Zi| is heavy-tailed The |Zi| are half-Cauchy 26 Proof of main theorem i |Zi| is heavy-tailed, so |RAx|1 = |Ax|1 i |Zi| / (d log d) may be large Each |Zi| has c.d.f. asymptotic to 1-Θ(1/z) for z in [0, 1) No problem! We know there exists a well-conditioned basis of A We can assume the basis vectors are A*1, …, A*d |RA*i|1 » |A*i|1 ¢ i |Zi| / (d log d) With constant probability, i |RA*i|1 = O(log d) i |A*i|1 27 Proof of main theorem Suppose i |RA*i|1 = O(log d) i |A*i|1 for well-conditioned basis A*1, …, A*d We will use the Auerbach basis which always exists: For all x, |x|1 · |Ax|1 i |A*i|1 = d I don’t know how to compute such a basis, but it doesn’t matter! i |RA*i|1 = O(d log d) |RAx|1 · i |RA*i xi| · |x|1 i |RA*i|1 = |x|1O(d log d) = O(d log d) |Ax|1 Q.E.D. 28 Main Theorem Theorem There is a probability space over (d log d) n matrices R such that for any nd matrix A, with probability at least 99/100 we have for all x: |Ax|1 ≤ |RAx|1 ≤ d log d ∙ |Ax|1 29 Outline Massive data sets Regression analysis Our results Our techniques Concluding remarks 30 Regression for data streams Streaming algorithm given additive updates to entries of A and b Pick random matrix R according to the distribution of main theorem Maintain RA and Rb during the stream Entries R do |RAx'-Rb|1 using linear programming Find x' thatof minimizes Compute U so not need tothat beRAU is orthonormal independent The hard thing is sampling rows from AU◦b’ proportional to their norm Do not know U, b’ until end of stream Surpisingly, there is still a way to do this in a single pass by treating U, x’ as formal variables and plugging them in at the end Uses a noisy sampling data structure Omitted from talk 31 Hyperplane Fitting Given n points in Rd, find hyperplane minimizing sum of l1-distances of points to the hyperplane Reduces to d invocations of l1-regression 32 Conclusion Main results Efficient algorithms for l1-regression and hyperplane fitting nd1.376 time improves previous nd5 log n running time for l1-regression First oblivious subspace embedding for l1 33