Learning Patterns for Detection with Multiscale Scan Statistics
|
|
- Neil Carroll
- 5 years ago
- Views:
Transcription
1 Learning Patterns for Detection with Multiscale Scan Statistics J. Sharpnack 1 1 Statistics Department UC Davis UC Davis Statistics Seminar 2018 Work supported by NSF DMS
2 Hyperspectral Gas Detection
3 Hyperspectral Gas Detection Each pixel of the image is a light spectrum, and we are interested in detecting a single chemical signature. Images from [Manolakis et al. 2014, Manolakis & Shaw 2002]
4 Hyperspectral Gas Detection Gas signature level in each pixel. Dataset and chemical signature comparison code from Dimitris G. Manolakis and DTRA.
5 Hyperspectral Gas Detection Decreased SNR by 1 5th. Can you see it?
6 Hyperspectral Gas Detection If classification answers the question, what am I seeing? detection answers the question, do I see anything at all?
7 Anomaly detection goals Goal 1: Reliably detect anomalous patterns within images beyond what the human eye can see at the precise information theoretic limit.
8 Detection applications Contaminant detection in water networks, real-time surveillance system, radiation monitoring, fire detection and other remote sensing applications, medical imaging and automated radiology, early detection of pathogen outbreaks.
9 Rectangular multiscale scan statistic Scan every rectangle (vary location and scale) looking for abnormal concentrations [S, Arias-Castro 16]. Figure: A rectangle over the active region.
10 Scan statistic Represent image over [ L, L] [ L, L] as matrix Y and consider scanning rectangle over pixels [ H 0, H 0 ] [ H 1, H 1 ]. Define the pattern to be P k,l = 1 (2H0 + 1) (2H 1 + 1) then the scan is the following convolution, (P Y ) k,l = H 0 k = H 0 H 1 l = H 1 Y k k,l l P k,l, where defined. The single-scale scan statistic is ŝ = max (P Y ) k,l. k,l
11 Other patterns General pattern over the domain Ω = [ L, L] d can be hidden in the noisy tensor. Figure: A simulated time series with an embedded sinusoidal signal with values on the y-axis (d = 1).
12 Multiscale scan Scanning with many pattern dimensions, H 0,..., H d, is called a multiscale scan. How do we compare scan statistics at different scales?
13 Anomaly detection goals Goal 2: General purpose analysis for multiscale scan statistics for a large class of patterns.
14 Multiple Tensors Figure: Multiple tensors, i = 1,..., n (n = 5), with different locations and scales, and two possible patterns (left and right).
15 Anomaly detection goals Goal 3: Leverage database of tensors to simulaneously learn and detect anomalous patterns.
16 Prior work [Naus 65] scan statistics introduced for point cloud data [Siegmund, Worsley 95] limit distribution of 1-dimensional scan [Glaz and Zhang 04, Kabluchko 11] limit in d-dimensions [Arias-Castro et al. 05, 11] scan for blob-like patterns [Dumbgen, Spokoiny, 01] scale adaptive scan statistic (d = 1) [S, Arias-Castro 16] scale adaptive rectangular scan [Proksch at al. 17] scale adaptive smooth patterns This work: learning and detecting general smooth patterns in a database of tensors with scale adaptive methods
17 Continuous model Pattern f F C 1 over [ 1, 1] d, f L2 = 1. Data is random measure dx i with domain [ L, L] d. Scale dilation f h := h 1/2 f (./h), h = j h j, h R d Null hypothesis: data is just noise (dw i is d-dimensional Wiener process) Alternative hypothesis: there is a signal f at location t i, and scale h i. H 0 : dx i (τ) = dw i (τ), i = 1,..., n H 1 : dx i (τ) = µf h i (t i τ)dτ + dw i (τ) for some f F, and (t i, h i ) D, i = 1,..., n.
18 Continuous multiscale scan statistic Convolution at scale h, (f h dx i )(t) = f h (τ)dx i (t τ) = 1 h f (τ)dx i (t hτ), Scale corrected multiscale scan statistic: s(x i ; f ) := max h H v h h H := j [1, L) t T h := j [ (L h j ), L h j ] v h = 2 j log(l/h j) ( max t T h (f h dx i )(t) v h ). (1) Test if the pattern f centered at t and scaled by h is hidden within tensor X i.
19 Learning patterns Given a dataset of images X i, i = 1,..., n then can we also learn the pattern f F? S n (X ; F) := max f F 1 n n s(x i ; f ) i=1 (PAMSS) The pattern adapted multiscale scan statistic (PAMSS) averages the MSS for each tensor. Smoothness conditions on F are required: bounded variation (TVC) or average Hölder condition (AHC).
20 Smoothness assumptions Assumption TVC: Define the isotropic total variation, f TV := f (u) 2 du, function are of bounded variation, Ω γ 1 > 0 s.t. f F, f TV γ 1. (TVC) or Assumption AHC: Define the Hölder functional, A t,s (f ) := f (t z) f (s z) 2 dz. Ω L functions have bounded average Hölder condition, 0 < γ 2 1 s.t. f F, A t,s (f ) c A t s 2γ 2 2. (AHC)
21 Type 1 error control We can simulate from the null distribution to obtain a significance level for ( ) s(x i ; f ) := max v h max(f h dx i )(t) v h. (2) h H t T h [Dumbgen & Spokoiny 01] showed that under H 0 for L large enough s(x i, f ) = O P (log log L) for functions satisfying (TVC) in 1D. We need a more precise control to analyze PAMSS (tail bound) s(x i, f ) is subexponential random variable.
22 SubGaussian process Definition We say that a random field, {Z(ι)} ι I, is a (zero mean) standard subgaussian process if there exists a constant u 0 > 0 such that P { Z(ι 0 ) Z(ι 1 ) u} 2 exp P {Z(ι 0 ) u} exp ( u2 2ν(ι 0, ι 1 ) ), (3) ) ( u2, (4) 2 for any ι 0, ι 1 I, u > u 0, and ν(ι 0, ι 1 ) = E(Z(ι 0 ) Z(ι 1 )) 2, is the canonical distance. Under H 0, {(f h dx i )(t) : (t, h) D} is a subgaussian random field with canonical distance ν f ((h 0, t 0 ), (h 1, t 1 )) := f h0 (t 0.) f h1 (t 1.) L2.
23 Dudley s chaining Define the covering number of metric space (I, ν), N (I, ν, ɛ), to be the number of balls of ν-radius ɛ that is required to cover I. Theorem (Dudley s entropy (tail) bound) { } P sup Z(η) > u c D + C η I Ce u2 2, such that D = 0 2 log N (I, ν, ɛ)dɛ for universal constants c, C. Specifically, if N (I, ν, ɛ) Γɛ ρ. then (as Γ ) D = 2 log Γ + o(1).
24 Dudley s chaining Find a sequence (the chain) {η k } k=0 such that lim k η k = η, sup Z(η) = sup Z(η 0 ) + (Z(η 1 ) Z(η 0 )) + (Z(η 2 ) Z(η 1 )) +... η I η 0,η 1,...
25 Dudley s chaining Find a sequence (the chain) {η k } k=0 such that lim k η k = η, sup Z(η) = sup Z(η 0 ) + (Z(η 1 ) Z(η 0 )) + (Z(η 2 ) Z(η 1 )) +... η I η 0,η 1,... Z(η i+1 ) Z(η i ) is prop. to ν(η i+1, η i ) Choose covering layers such that N grows like 2 2k Variance of the bound is dominated by Z(η 0 )
26 Chaining standardized suprema Theorem Let Z(η) be a standard subgaussian process over an index set I. Suppose that metric (I, d Z ) has covering number, N s.t., N (I, d Z, ɛ) Γɛ ρ. (5) Then there exists an Γ 0 > 0 such that for any Γ Γ 0, the following supremum is bounded in probability, ( ) } { c0 P log Γ sup Z(η) 2 log Γ a 0 log log Γ > u e u, η I (6) for u > u 0 where u 0, c 0, a 0 are constant depending on ρ (but not on Γ). In words, the supremum of such a subgaussian process is subexponential with location and rate parameter, (2 log Γ) 1/2 (omitting the log log term).
27 Proof sketch for chaining bound For iid normals, {z i } N i=1, from union bound { P max z i > } 2 log N + u 2 e u2 2. i Generic chaining: 2 log N + u 2 u + 2 log N/(2u) Our chaining: 2 log N + u 2 2 log N + u 2 /(2 2 log N) { P 2 ( 2 log N max z i ) 2 log N i Modify chain so that } > u e u. (1) start the chain at a deeper level (N large enough) (2) make the covers grow slowly N a ak for a 1.
28 Main Theorem Theorem Let F be finite and assume that either all functions in F satisfy either (TVC) or (AHC). Let K log F n (δ) := ( F δ K n log ( F δ then for some constant K, under H 0, (7) ) ), log F n K + log δ, log F > n K + log δ P {S n (X, F) > F n (δ) log log L} δ. (8)
29 Proof of main theorem Lemma Then, under the above conditions, there is a constant C depending on d alone such that 1. Suppose that (TVC) holds for the class F, then ν f ((t, h), (t, h )) 2 Cγ 1 t t 2 ( ) 2 h h + 1. h 2. [Proksch et al. 16] Suppose that (AHC) holds for the class F, then ν f ((t, h), (t, h )) 2 t j t 2γ 2 j h C + j h 2γ 2 j h j 2γ 2 h j h j. 2 2γ 2
30 Proof of main theorem Lemma Suppose that f F satisfies either (TVC) or (AHC). Let l {0,..., log 2 L } d, and H 2 (l) = j [2 l j, 2 l j +1 ]. Then { } ( P c 1 max v h (fh dx i ) )(t) v h a1 log log L > u e u h H l,t T h for constants a 1, c 1 > 0 depending on γ, d only. With the union bound over l, { P c 2 sn(x i }, f ) log log L a 2 > u e u, then use subexponential Bernstein inequality.
31 Asymptotic Distinguishability Corollary Suppose that log F = o(n), and recall that under the alternative hypothesis, H 1, X i has an embedded pattern f at scale h i and v 2 = h i j log(l/hi j ), and the noise is a standard Wiener process. Suppose also that hj i Lc for some 0 c < 1 for all i, j, then the PAMSS is asymptotically powerful (has diminishing probability of type 1 and type 2 error) if µ 2 n i=1 v 2 h i n i=1 v h i. (9) We take this result to mean that as long as the function class, F, does not grow exponentially in n, we achieve asymptotic power under the same conditions as if F = 1.
32 Summary Proposed the Pattern Adapted Multiscale Scan Statistic for learning anomalous patterns We proved a refined concentration result for the supremum of subgaussian processes We controlled the error probabilities for the PAMSS showing that it can learn the function from a class for free as long as n log F For a link to this paper: Thanks!
Information theoretic perspectives on learning algorithms
Information theoretic perspectives on learning algorithms Varun Jog University of Wisconsin - Madison Departments of ECE and Mathematics Shannon Channel Hangout! May 8, 2018 Jointly with Adrian Tovar-Lopez
More informationRANDOM FIELDS AND GEOMETRY. Robert Adler and Jonathan Taylor
RANDOM FIELDS AND GEOMETRY from the book of the same name by Robert Adler and Jonathan Taylor IE&M, Technion, Israel, Statistics, Stanford, US. ie.technion.ac.il/adler.phtml www-stat.stanford.edu/ jtaylor
More informationSupplementary Material for Nonparametric Operator-Regularized Covariance Function Estimation for Functional Data
Supplementary Material for Nonparametric Operator-Regularized Covariance Function Estimation for Functional Data Raymond K. W. Wong Department of Statistics, Texas A&M University Xiaoke Zhang Department
More informationPoisson Approximation for Two Scan Statistics with Rates of Convergence
Poisson Approximation for Two Scan Statistics with Rates of Convergence Xiao Fang (Joint work with David Siegmund) National University of Singapore May 28, 2015 Outline The first scan statistic The second
More informationClass 2 & 3 Overfitting & Regularization
Class 2 & 3 Overfitting & Regularization Carlo Ciliberto Department of Computer Science, UCL October 18, 2017 Last Class The goal of Statistical Learning Theory is to find a good estimator f n : X Y, approximating
More informationGeometric inference for measures based on distance functions The DTM-signature for a geometric comparison of metric-measure spaces from samples
Distance to Geometric inference for measures based on distance functions The - for a geometric comparison of metric-measure spaces from samples the Ohio State University wan.252@osu.edu 1/31 Geometric
More informationIntroduction to Empirical Processes and Semiparametric Inference Lecture 12: Glivenko-Cantelli and Donsker Results
Introduction to Empirical Processes and Semiparametric Inference Lecture 12: Glivenko-Cantelli and Donsker Results Michael R. Kosorok, Ph.D. Professor and Chair of Biostatistics Professor of Statistics
More informationg(x) = P (y) Proof. This is true for n = 0. Assume by the inductive hypothesis that g (n) (0) = 0 for some n. Compute g (n) (h) g (n) (0)
Mollifiers and Smooth Functions We say a function f from C is C (or simply smooth) if all its derivatives to every order exist at every point of. For f : C, we say f is C if all partial derivatives to
More informationSupremum of simple stochastic processes
Subspace embeddings Daniel Hsu COMS 4772 1 Supremum of simple stochastic processes 2 Recap: JL lemma JL lemma. For any ε (0, 1/2), point set S R d of cardinality 16 ln n S = n, and k N such that k, there
More informationConcentration behavior of the penalized least squares estimator
Concentration behavior of the penalized least squares estimator Penalized least squares behavior arxiv:1511.08698v2 [math.st] 19 Oct 2016 Alan Muro and Sara van de Geer {muro,geer}@stat.math.ethz.ch Seminar
More informationGaussian Random Fields: Excursion Probabilities
Gaussian Random Fields: Excursion Probabilities Yimin Xiao Michigan State University Lecture 5 Excursion Probabilities 1 Some classical results on excursion probabilities A large deviation result Upper
More informationBrownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539
Brownian motion Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory
More informationTools from Lebesgue integration
Tools from Lebesgue integration E.P. van den Ban Fall 2005 Introduction In these notes we describe some of the basic tools from the theory of Lebesgue integration. Definitions and results will be given
More informationIntroduction to Empirical Processes and Semiparametric Inference Lecture 09: Stochastic Convergence, Continued
Introduction to Empirical Processes and Semiparametric Inference Lecture 09: Stochastic Convergence, Continued Michael R. Kosorok, Ph.D. Professor and Chair of Biostatistics Professor of Statistics and
More informationConsistency of Modularity Clustering on Random Geometric Graphs
Consistency of Modularity Clustering on Random Geometric Graphs Erik Davis The University of Arizona May 10, 2016 Outline Introduction to Modularity Clustering Pointwise Convergence Convergence of Optimal
More informationCourse 212: Academic Year Section 1: Metric Spaces
Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........
More informationInformation-Theoretic Limits of Group Testing: Phase Transitions, Noisy Tests, and Partial Recovery
Information-Theoretic Limits of Group Testing: Phase Transitions, Noisy Tests, and Partial Recovery Jonathan Scarlett jonathan.scarlett@epfl.ch Laboratory for Information and Inference Systems (LIONS)
More informationL p Functions. Given a measure space (X, µ) and a real number p [1, ), recall that the L p -norm of a measurable function f : X R is defined by
L p Functions Given a measure space (, µ) and a real number p [, ), recall that the L p -norm of a measurable function f : R is defined by f p = ( ) /p f p dµ Note that the L p -norm of a function f may
More informationProblem set 1, Real Analysis I, Spring, 2015.
Problem set 1, Real Analysis I, Spring, 015. (1) Let f n : D R be a sequence of functions with domain D R n. Recall that f n f uniformly if and only if for all ɛ > 0, there is an N = N(ɛ) so that if n
More informationl 1 -Regularized Linear Regression: Persistence and Oracle Inequalities
l -Regularized Linear Regression: Persistence and Oracle Inequalities Peter Bartlett EECS and Statistics UC Berkeley slides at http://www.stat.berkeley.edu/ bartlett Joint work with Shahar Mendelson and
More informationMath 127C, Spring 2006 Final Exam Solutions. x 2 ), g(y 1, y 2 ) = ( y 1 y 2, y1 2 + y2) 2. (g f) (0) = g (f(0))f (0).
Math 27C, Spring 26 Final Exam Solutions. Define f : R 2 R 2 and g : R 2 R 2 by f(x, x 2 (sin x 2 x, e x x 2, g(y, y 2 ( y y 2, y 2 + y2 2. Use the chain rule to compute the matrix of (g f (,. By the chain
More informationAsymptotic Nonequivalence of Nonparametric Experiments When the Smoothness Index is ½
University of Pennsylvania ScholarlyCommons Statistics Papers Wharton Faculty Research 1998 Asymptotic Nonequivalence of Nonparametric Experiments When the Smoothness Index is ½ Lawrence D. Brown University
More informationTesting Algebraic Hypotheses
Testing Algebraic Hypotheses Mathias Drton Department of Statistics University of Chicago 1 / 18 Example: Factor analysis Multivariate normal model based on conditional independence given hidden variable:
More informationIntegral representations in models with long memory
Integral representations in models with long memory Georgiy Shevchenko, Yuliya Mishura, Esko Valkeila, Lauri Viitasaari, Taras Shalaiko Taras Shevchenko National University of Kyiv 29 September 215, Ulm
More informationLecture No 1 Introduction to Diffusion equations The heat equat
Lecture No 1 Introduction to Diffusion equations The heat equation Columbia University IAS summer program June, 2009 Outline of the lectures We will discuss some basic models of diffusion equations and
More informationThe geometry of Gaussian processes and Bayesian optimization. Contal CMLA, ENS Cachan
The geometry of Gaussian processes and Bayesian optimization. Contal CMLA, ENS Cachan Background: Global Optimization and Gaussian Processes The Geometry of Gaussian Processes and the Chaining Trick Algorithm
More informationPCA with random noise. Van Ha Vu. Department of Mathematics Yale University
PCA with random noise Van Ha Vu Department of Mathematics Yale University An important problem that appears in various areas of applied mathematics (in particular statistics, computer science and numerical
More informationFast learning rates for plug-in classifiers under the margin condition
Fast learning rates for plug-in classifiers under the margin condition Jean-Yves Audibert 1 Alexandre B. Tsybakov 2 1 Certis ParisTech - Ecole des Ponts, France 2 LPMA Université Pierre et Marie Curie,
More informationNotes on Distributions
Notes on Distributions Functional Analysis 1 Locally Convex Spaces Definition 1. A vector space (over R or C) is said to be a topological vector space (TVS) if it is a Hausdorff topological space and the
More informationCan we do statistical inference in a non-asymptotic way? 1
Can we do statistical inference in a non-asymptotic way? 1 Guang Cheng 2 Statistics@Purdue www.science.purdue.edu/bigdata/ ONR Review Meeting@Duke Oct 11, 2017 1 Acknowledge NSF, ONR and Simons Foundation.
More informationLECTURE 2: LOCAL TIME FOR BROWNIAN MOTION
LECTURE 2: LOCAL TIME FOR BROWNIAN MOTION We will define local time for one-dimensional Brownian motion, and deduce some of its properties. We will then use the generalized Ray-Knight theorem proved in
More informationsample lectures: Compressed Sensing, Random Sampling I & II
P. Jung, CommIT, TU-Berlin]: sample lectures Compressed Sensing / Random Sampling I & II 1/68 sample lectures: Compressed Sensing, Random Sampling I & II Peter Jung Communications and Information Theory
More informationON CONVERGENCE RATES OF GIBBS SAMPLERS FOR UNIFORM DISTRIBUTIONS
The Annals of Applied Probability 1998, Vol. 8, No. 4, 1291 1302 ON CONVERGENCE RATES OF GIBBS SAMPLERS FOR UNIFORM DISTRIBUTIONS By Gareth O. Roberts 1 and Jeffrey S. Rosenthal 2 University of Cambridge
More informationLecture 21: Minimax Theory
Lecture : Minimax Theory Akshay Krishnamurthy akshay@cs.umass.edu November 8, 07 Recap In the first part of the course, we spent the majority of our time studying risk minimization. We found many ways
More informationTail inequalities for additive functionals and empirical processes of. Markov chains
Tail inequalities for additive functionals and empirical processes of geometrically ergodic Markov chains University of Warsaw Banff, June 2009 Geometric ergodicity Definition A Markov chain X = (X n )
More informationDECOUPLING LECTURE 6
18.118 DECOUPLING LECTURE 6 INSTRUCTOR: LARRY GUTH TRANSCRIBED BY DONGHAO WANG We begin by recalling basic settings of multi-linear restriction problem. Suppose Σ i,, Σ n are some C 2 hyper-surfaces in
More informationASYMPTOTIC EQUIVALENCE OF DENSITY ESTIMATION AND GAUSSIAN WHITE NOISE. By Michael Nussbaum Weierstrass Institute, Berlin
The Annals of Statistics 1996, Vol. 4, No. 6, 399 430 ASYMPTOTIC EQUIVALENCE OF DENSITY ESTIMATION AND GAUSSIAN WHITE NOISE By Michael Nussbaum Weierstrass Institute, Berlin Signal recovery in Gaussian
More informationNonparametric regression with martingale increment errors
S. Gaïffas (LSTA - Paris 6) joint work with S. Delattre (LPMA - Paris 7) work in progress Motivations Some facts: Theoretical study of statistical algorithms requires stationary and ergodicity. Concentration
More informationEmpirical Processes and random projections
Empirical Processes and random projections B. Klartag, S. Mendelson School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA. Institute of Advanced Studies, The Australian National
More informationLearning Theory. Ingo Steinwart University of Stuttgart. September 4, 2013
Learning Theory Ingo Steinwart University of Stuttgart September 4, 2013 Ingo Steinwart University of Stuttgart () Learning Theory September 4, 2013 1 / 62 Basics Informal Introduction Informal Description
More informationChapter 9: Basic of Hypercontractivity
Analysis of Boolean Functions Prof. Ryan O Donnell Chapter 9: Basic of Hypercontractivity Date: May 6, 2017 Student: Chi-Ning Chou Index Problem Progress 1 Exercise 9.3 (Tightness of Bonami Lemma) 2/2
More informationMATH MEASURE THEORY AND FOURIER ANALYSIS. Contents
MATH 3969 - MEASURE THEORY AND FOURIER ANALYSIS ANDREW TULLOCH Contents 1. Measure Theory 2 1.1. Properties of Measures 3 1.2. Constructing σ-algebras and measures 3 1.3. Properties of the Lebesgue measure
More informationOn the Complexity of Best Arm Identification with Fixed Confidence
On the Complexity of Best Arm Identification with Fixed Confidence Discrete Optimization with Noise Aurélien Garivier, Emilie Kaufmann COLT, June 23 th 2016, New York Institut de Mathématiques de Toulouse
More informationSobolev Spaces. Chapter 10
Chapter 1 Sobolev Spaces We now define spaces H 1,p (R n ), known as Sobolev spaces. For u to belong to H 1,p (R n ), we require that u L p (R n ) and that u have weak derivatives of first order in L p
More informationPlug-in Approach to Active Learning
Plug-in Approach to Active Learning Stanislav Minsker Stanislav Minsker (Georgia Tech) Plug-in approach to active learning 1 / 18 Prediction framework Let (X, Y ) be a random couple in R d { 1, +1}. X
More informationSTAT 200C: High-dimensional Statistics
STAT 200C: High-dimensional Statistics Arash A. Amini May 30, 2018 1 / 57 Table of Contents 1 Sparse linear models Basis Pursuit and restricted null space property Sufficient conditions for RNS 2 / 57
More information21.2 Example 1 : Non-parametric regression in Mean Integrated Square Error Density Estimation (L 2 2 risk)
10-704: Information Processing and Learning Spring 2015 Lecture 21: Examples of Lower Bounds and Assouad s Method Lecturer: Akshay Krishnamurthy Scribes: Soumya Batra Note: LaTeX template courtesy of UC
More informationRisk Bounds for Lévy Processes in the PAC-Learning Framework
Risk Bounds for Lévy Processes in the PAC-Learning Framework Chao Zhang School of Computer Engineering anyang Technological University Dacheng Tao School of Computer Engineering anyang Technological University
More informationAsymptotic results for empirical measures of weighted sums of independent random variables
Asymptotic results for empirical measures of weighted sums of independent random variables B. Bercu and W. Bryc University Bordeaux 1, France Workshop on Limit Theorems, University Paris 1 Paris, January
More informationLaplace s Equation. Chapter Mean Value Formulas
Chapter 1 Laplace s Equation Let be an open set in R n. A function u C 2 () is called harmonic in if it satisfies Laplace s equation n (1.1) u := D ii u = 0 in. i=1 A function u C 2 () is called subharmonic
More informationGeneralization Bounds in Machine Learning. Presented by: Afshin Rostamizadeh
Generalization Bounds in Machine Learning Presented by: Afshin Rostamizadeh Outline Introduction to generalization bounds. Examples: VC-bounds Covering Number bounds Rademacher bounds Stability bounds
More informationCombinatorial Variants of Lebesgue s Density Theorem
Combinatorial Variants of Lebesgue s Density Theorem Philipp Schlicht joint with David Schrittesser, Sandra Uhlenbrock and Thilo Weinert September 6, 2017 Philipp Schlicht Variants of Lebesgue s Density
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationPhenomena in high dimensions in geometric analysis, random matrices, and computational geometry Roscoff, France, June 25-29, 2012
Phenomena in high dimensions in geometric analysis, random matrices, and computational geometry Roscoff, France, June 25-29, 202 BOUNDS AND ASYMPTOTICS FOR FISHER INFORMATION IN THE CENTRAL LIMIT THEOREM
More informationHölder continuity of the solution to the 3-dimensional stochastic wave equation
Hölder continuity of the solution to the 3-dimensional stochastic wave equation (joint work with Yaozhong Hu and Jingyu Huang) Department of Mathematics Kansas University CBMS Conference: Analysis of Stochastic
More informationu( x) = g( y) ds y ( 1 ) U solves u = 0 in U; u = 0 on U. ( 3)
M ath 5 2 7 Fall 2 0 0 9 L ecture 4 ( S ep. 6, 2 0 0 9 ) Properties and Estimates of Laplace s and Poisson s Equations In our last lecture we derived the formulas for the solutions of Poisson s equation
More informationGaussian Random Fields: Geometric Properties and Extremes
Gaussian Random Fields: Geometric Properties and Extremes Yimin Xiao Michigan State University Outline Lecture 1: Gaussian random fields and their regularity Lecture 2: Hausdorff dimension results and
More informationInference for High Dimensional Robust Regression
Department of Statistics UC Berkeley Stanford-Berkeley Joint Colloquium, 2015 Table of Contents 1 Background 2 Main Results 3 OLS: A Motivating Example Table of Contents 1 Background 2 Main Results 3 OLS:
More informationMARKOV PARTITIONS FOR HYPERBOLIC SETS
MARKOV PARTITIONS FOR HYPERBOLIC SETS TODD FISHER, HIMAL RATHNAKUMARA Abstract. We show that if f is a diffeomorphism of a manifold to itself, Λ is a mixing (or transitive) hyperbolic set, and V is a neighborhood
More information1 Lesson 1: Brunn Minkowski Inequality
1 Lesson 1: Brunn Minkowski Inequality A set A R n is called convex if (1 λ)x + λy A for any x, y A and any λ [0, 1]. The Minkowski sum of two sets A, B R n is defined by A + B := {a + b : a A, b B}. One
More information212a1416 Wiener measure.
212a1416 Wiener measure. November 11, 2014 1 Wiener measure. A refined version of the Riesz representation theorem for measures The Big Path Space. The heat equation. Paths are continuous with probability
More informationAnomalous transport of particles in Plasma physics
Anomalous transport of particles in Plasma physics L. Cesbron a, A. Mellet b,1, K. Trivisa b, a École Normale Supérieure de Cachan Campus de Ker Lann 35170 Bruz rance. b Department of Mathematics, University
More informationLarge-Scale L1-Related Minimization in Compressive Sensing and Beyond
Large-Scale L1-Related Minimization in Compressive Sensing and Beyond Yin Zhang Department of Computational and Applied Mathematics Rice University, Houston, Texas, U.S.A. Arizona State University March
More information1 Regression with High Dimensional Data
6.883 Learning with Combinatorial Structure ote for Lecture 11 Instructor: Prof. Stefanie Jegelka Scribe: Xuhong Zhang 1 Regression with High Dimensional Data Consider the following regression problem:
More informationEntropy, Inference, and Channel Coding
Entropy, Inference, and Channel Coding Sean Meyn Department of Electrical and Computer Engineering University of Illinois and the Coordinated Science Laboratory NSF support: ECS 02-17836, ITR 00-85929
More informationLecture 2: Uniform Entropy
STAT 583: Advanced Theory of Statistical Inference Spring 218 Lecture 2: Uniform Entropy Lecturer: Fang Han April 16 Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal
More informationMATHS 730 FC Lecture Notes March 5, Introduction
1 INTRODUCTION MATHS 730 FC Lecture Notes March 5, 2014 1 Introduction Definition. If A, B are sets and there exists a bijection A B, they have the same cardinality, which we write as A, #A. If there exists
More informationAsymptotic results for empirical measures of weighted sums of independent random variables
Asymptotic results for empirical measures of weighted sums of independent random variables B. Bercu and W. Bryc University Bordeaux 1, France Seminario di Probabilità e Statistica Matematica Sapienza Università
More information(1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define
Homework, Real Analysis I, Fall, 2010. (1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define ρ(f, g) = 1 0 f(x) g(x) dx. Show that
More informationRate-Optimal Detection of Very Short Signal Segments
Rate-Optimal Detection of Very Short Signal Segments T. Tony Cai and Ming Yuan University of Pennsylvania and University of Wisconsin-Madison July 6, 2014) Abstract Motivated by a range of applications
More informationConsistency of the maximum likelihood estimator for general hidden Markov models
Consistency of the maximum likelihood estimator for general hidden Markov models Jimmy Olsson Centre for Mathematical Sciences Lund University Nordstat 2012 Umeå, Sweden Collaborators Hidden Markov models
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University The Residual and Error of Finite Element Solutions Mixed BVP of Poisson Equation
More informationEECS 750. Hypothesis Testing with Communication Constraints
EECS 750 Hypothesis Testing with Communication Constraints Name: Dinesh Krithivasan Abstract In this report, we study a modification of the classical statistical problem of bivariate hypothesis testing.
More informationBrownian Motion. 1 Definition Brownian Motion Wiener measure... 3
Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................
More informationCones of measures. Tatiana Toro. University of Washington. Quantitative and Computational Aspects of Metric Geometry
Cones of measures Tatiana Toro University of Washington Quantitative and Computational Aspects of Metric Geometry Based on joint work with C. Kenig and D. Preiss Tatiana Toro (University of Washington)
More informationLetting p shows that {B t } t 0. Definition 0.5. For λ R let δ λ : A (V ) A (V ) be defined by. 1 = g (symmetric), and. 3. g
4 Contents.1 Lie group p variation results Suppose G, d) is a group equipped with a left invariant metric, i.e. Let a := d e, a), then d ca, cb) = d a, b) for all a, b, c G. d a, b) = d e, a 1 b ) = a
More informationAdditive functionals of infinite-variance moving averages. Wei Biao Wu The University of Chicago TECHNICAL REPORT NO. 535
Additive functionals of infinite-variance moving averages Wei Biao Wu The University of Chicago TECHNICAL REPORT NO. 535 Departments of Statistics The University of Chicago Chicago, Illinois 60637 June
More informationApproximating the Integrated Tail Distribution
Approximating the Integrated Tail Distribution Ants Kaasik & Kalev Pärna University of Tartu VIII Tartu Conference on Multivariate Statistics 26th of June, 2007 Motivating example Let W be a steady-state
More informationSmall ball probabilities and metric entropy
Small ball probabilities and metric entropy Frank Aurzada, TU Berlin Sydney, February 2012 MCQMC Outline 1 Small ball probabilities vs. metric entropy 2 Connection to other questions 3 Recent results for
More informationA note on some approximation theorems in measure theory
A note on some approximation theorems in measure theory S. Kesavan and M. T. Nair Department of Mathematics, Indian Institute of Technology, Madras, Chennai - 600 06 email: kesh@iitm.ac.in and mtnair@iitm.ac.in
More informationTHE LASSO, CORRELATED DESIGN, AND IMPROVED ORACLE INEQUALITIES. By Sara van de Geer and Johannes Lederer. ETH Zürich
Submitted to the Annals of Applied Statistics arxiv: math.pr/0000000 THE LASSO, CORRELATED DESIGN, AND IMPROVED ORACLE INEQUALITIES By Sara van de Geer and Johannes Lederer ETH Zürich We study high-dimensional
More informationValuations. 6.1 Definitions. Chapter 6
Chapter 6 Valuations In this chapter, we generalize the notion of absolute value. In particular, we will show how the p-adic absolute value defined in the previous chapter for Q can be extended to hold
More information10-704: Information Processing and Learning Fall Lecture 21: Nov 14. sup
0-704: Information Processing and Learning Fall 206 Lecturer: Aarti Singh Lecture 2: Nov 4 Note: hese notes are based on scribed notes from Spring5 offering of this course LaeX template courtesy of UC
More informationMATH Final Project Mean Curvature Flows
MATH 581 - Final Project Mean Curvature Flows Olivier Mercier April 30, 2012 1 Introduction The mean curvature flow is part of the bigger family of geometric flows, which are flows on a manifold associated
More informationfor all subintervals I J. If the same is true for the dyadic subintervals I D J only, we will write ϕ BMO d (J). In fact, the following is true
3 ohn Nirenberg inequality, Part I A function ϕ L () belongs to the space BMO() if sup ϕ(s) ϕ I I I < for all subintervals I If the same is true for the dyadic subintervals I D only, we will write ϕ BMO
More informationQuasi-conformal maps and Beltrami equation
Chapter 7 Quasi-conformal maps and Beltrami equation 7. Linear distortion Assume that f(x + iy) =u(x + iy)+iv(x + iy) be a (real) linear map from C C that is orientation preserving. Let z = x + iy and
More informationEmpirical Risk Minimization
Empirical Risk Minimization Fabrice Rossi SAMM Université Paris 1 Panthéon Sorbonne 2018 Outline Introduction PAC learning ERM in practice 2 General setting Data X the input space and Y the output space
More informationRigidity and Non-rigidity Results on the Sphere
Rigidity and Non-rigidity Results on the Sphere Fengbo Hang Xiaodong Wang Department of Mathematics Michigan State University Oct., 00 1 Introduction It is a simple consequence of the maximum principle
More informationBennett-type Generalization Bounds: Large-deviation Case and Faster Rate of Convergence
Bennett-type Generalization Bounds: Large-deviation Case and Faster Rate of Convergence Chao Zhang The Biodesign Institute Arizona State University Tempe, AZ 8587, USA Abstract In this paper, we present
More informationMixing time for a random walk on a ring
Mixing time for a random walk on a ring Stephen Connor Joint work with Michael Bate Paris, September 2013 Introduction Let X be a discrete time Markov chain on a finite state space S, with transition matrix
More informationConcentration Inequalities
Chapter Concentration Inequalities I. Moment generating functions, the Chernoff method, and sub-gaussian and sub-exponential random variables a. Goal for this section: given a random variable X, how does
More informationSome functional (Hölderian) limit theorems and their applications (II)
Some functional (Hölderian) limit theorems and their applications (II) Alfredas Račkauskas Vilnius University Outils Statistiques et Probabilistes pour la Finance Université de Rouen June 1 5, Rouen (Rouen
More informationA Note on the Central Limit Theorem for a Class of Linear Systems 1
A Note on the Central Limit Theorem for a Class of Linear Systems 1 Contents Yukio Nagahata Department of Mathematics, Graduate School of Engineering Science Osaka University, Toyonaka 560-8531, Japan.
More information3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable
More informationFunctional Analysis Exercise Class
Functional Analysis Exercise Class Week 2 November 6 November Deadline to hand in the homeworks: your exercise class on week 9 November 13 November Exercises (1) Let X be the following space of piecewise
More informationUpper Bound for Intermediate Singular Values of Random Sub-Gaussian Matrices 1
Upper Bound for Intermediate Singular Values of Random Sub-Gaussian Matrices 1 Feng Wei 2 University of Michigan July 29, 2016 1 This presentation is based a project under the supervision of M. Rudelson.
More informationCovariance function estimation in Gaussian process regression
Covariance function estimation in Gaussian process regression François Bachoc Department of Statistics and Operations Research, University of Vienna WU Research Seminar - May 2015 François Bachoc Gaussian
More informationbe the set of complex valued 2π-periodic functions f on R such that
. Fourier series. Definition.. Given a real number P, we say a complex valued function f on R is P -periodic if f(x + P ) f(x) for all x R. We let be the set of complex valued -periodic functions f on
More informationMath 6455 Nov 1, Differential Geometry I Fall 2006, Georgia Tech
Math 6455 Nov 1, 26 1 Differential Geometry I Fall 26, Georgia Tech Lecture Notes 14 Connections Suppose that we have a vector field X on a Riemannian manifold M. How can we measure how much X is changing
More information