Sparse Nonparametric Density Estimation in High Dimensions Using the Rodeo

Transcription

1 Outline in High Dimensions Using the Rodeo Han Liu 1,2 John Lafferty 2,3 Larry Wasserman 1,2 1 Statistics Department, 2 Machine Learning Department, 3 Computer Science Department, Carnegie Mellon University July 1st, 2006

2 Outline Motivation Research background Rodeo is a general strategy for nonparametric inference. It has been successfully applied to solve sparse nonparametric regression problems in high dimensions by Lafferty & Wasserman, Our goal Trying to adapt the rodeo framework to nonparametric density estimation problems. So that we have a unified framework for both density estimation and regression problems which is computationally efficient and theoretically soundable

3 Outline Outline 1 Background Nonparametric density estimation in high dimensions Sparsity assumptions for density estimation 2 The main idea The local rodeo algorithm for the kernel density estimator 3 The asymptotic running time and minimax risk 4 The global density rodeo and the reverse density rodeo Using other distributions as irrelevant dimensions 5 Empirical results on both synthetic and real-world datasets

4 Problem statement Nonparametric density estimation in high dimensions Sparsity assumptions and the rodeo framework Problem To estimate the joint density of a continuous d-dimensional random vector X = (X 1, X 2,..., X d ) F, d 3 where F is the unknown distribution with density function f(x). This problem is essentially hard, since the high dimensionality causes both computational and theoretical problems.

5 Previous work Background Nonparametric density estimation in high dimensions Sparsity assumptions and the rodeo framework From a frequentist perspective Kernel density estimation and the local likelihood method Projection pursuit method Log-spline models and the penalized likelihood method From a Bayesian perspective Mixture of normals with Dirichlet processes as prior Difficulties of current approaches Some methods only work well for low-dimensional problems Some heuristics lack the theoretical guarantees More importantly, they suffer from the curse of dimensionality

6 The curse of dimensionality Nonparametric density estimation in high dimensions Sparsity assumptions and the rodeo framework Characterizing the curse In a Sobolev space of order k, minimax theory shows that the best convergence rate for the mean squared error is R opt = O (n 2k/(2k+d)) which is practically slow when the dimension d is large. Combating the curse by some sparsity assumptions If the high-dimensional data has a low dimensional structure or a sparsity condition, we expect that some methods could be developed to combat the curse of dimensionality. This motivates the development of the rodeo framework

7 Rodeo for nonparametric regression (I) Nonparametric density estimation in high dimensions Sparsity assumptions and the rodeo framework Rodeo (regularization of derivative expectation operator) is a general strategy for nonparametric inference. Which has been used for nonparametric regression For a regression problem Y i = m(x i ) + ɛ i, i = 1,..., n where X i = (X i1,..., X id ) R d is a d-dimensional covariate. If m is in a d-dimensional Sobolev space of order 2, the best convergence rate for the risk is R = O (n 4/(4+d)) Which shows the curse of dimensionality in a regression setting.

8 Rodeo for nonparametric regression (II) Nonparametric density estimation in high dimensions Sparsity assumptions and the rodeo framework Assume the true function only depends on r covariates (r d) m(x) = m(x 1,..., x r ) for any ɛ > 0, the rodeo can simultaneously perform bandwidth selection and (implicitly) variable selection to achieve a better minimax convergence rate of R rodeo = O (n 4/(4+r)+ɛ) as if the r relevant variables were explicitly isolated in advance. Rodeo beats the curse of dimensionality in this sense. We expect to apply the same idea to solve density estimation problems.

9 Sparse density estimation Nonparametric density estimation in high dimensions Sparsity assumptions and the rodeo framework For many applications, the true density function can be characterized by some low dimensional structure Sparsity assumption for density estimation problems Assume h jj (x) is the second partial derivative of h on the j-th varaible, there exists some r d, such that f(x) g(x 1,..., x r )h(x) where h jj (x) = 0 for j = 1,..., d. Where x R = {x 1,..., x r } are the relevant dimensions. This condition imposes that h( ) belongs to a family of very smooth functions (e.g. the uniform distribution). h( ) can be generalized to be any parametric distribution!

10 Generalized sparse density estimation Nonparametric density estimation in high dimensions Sparsity assumptions and the rodeo framework We can generalize h( ) to other distributions (e.g. Gaussian). General sparsity assumption for density estimation problems Assume h( ) is any distribution (e.g. Gaussian) that we are not interested in f(x) g(x 1,..., x r )h(x) where r d. Thus, the density function f( ) can be factored into two parts: the relevant components g( ) and the irrelevant components h( ). Where x R = {x 1,..., x r } are the relevant dimensions. Under this framework, we can hope to achieve a better minimax rate R rodeo = O (n 4/(4+r))

11 Related work Background Nonparametric density estimation in high dimensions Sparsity assumptions and the rodeo framework Recent work that addressed this problem Minimum volume set (Scott& Nowak, JMLR06) Nongaussian component analysis; (Blanchard et al. JMLR06) Log-ANOVA model; (Lin & Joen, Statistical Sinica 2006) Advantages of our approach: Rodeo can utilize well-established nonparametric estimators A unified framework for different kinds of problems easy to implement and is amenable to theoretical analysis

12 Density rodeo: the main idea The main idea The local rodeo algorithm for the kernel density estim The key intuition: if a dimension is irrelevant, then changing the smoothing parameter of that dimension should only result in a small change in the whole estimator Basically, Rodeo is just a regularization strategy Use a kernel density estimator start with large bandwidths Calculate the gradient of the estimator w.r.t. the bandwidth Sequentially decrease the bandwidths in a greedy way, and try to freeze this decay process by some thresholding strategy to achieve a sparse solution

13 Density rodeo: the main idea The main idea The local rodeo algorithm for the kernel density estim Assuming a fixed point x and let ˆf H (x) denote an estimator of f(x) based on smoothing parameter matrix H = diag(h 1,..., h d ) Let M(h) = E( ˆf h (x)) denote the mean of ˆf h (x), therefore, f(x) = M(0) = E( ˆf 0 (x)). Assuming P = {h(t) : 0 t 1} is a smooth path through the set of smoothing parameters with h(0) = 0 and h(1) = 1, then 1 f(x) = M(1) (M(1) M(0)) = M(1) where D(h) = M(h) = ḣ(s) = dh(s) ds = M(1) dm(h(s)) ds ds D(s), ḣ(s) ds ( M h 1,..., M h d ) T is the gradient of M(h) and is the derivative of h(s) along the path.

14 Density rodeo: the main idea The main idea The local rodeo algorithm for the kernel density estim A biased, low variance estimator of M(1) is ˆf 1 (x). An unbiased estimator of D(h) is Z(h) = ( ˆf H (x),..., ˆf ) T H (x) h 1 h d This naive estimator has poor risk due to the large variance of Z(h) for small bandwidth. However, the sparsity assumption on f suggests that there should be paths for which D(h) is also sparse. Along such a path, Z(h) could be replaced with an estimator ˆD(h) that makes use of the sparsity assumption. Then, the estimate of f(x) becomes f H (x) = ˆf 1 (x) 1 0 ˆD(s), ḣ(s) ds

15 Kernel density estimator The main idea The local rodeo algorithm for the kernel density estim Let ˆf H (x) represents the kernel density estimator of f(x) with a bandwidth matrix H. assuming that K is a standard symmetric kernel, s.t. K(u)du = 1, uk(u)du = 0 d while K H ( ) = 1 det(h) K(H 1 ) represents the kernel with bandwidth matrix H = diag(h 1,..., h d ). ˆf H (x) = = 1 n 1 n det(h) n d i=i j=1 n K(H 1 (x X i )) i=1 1 h j K Here, we assume that K is a product kernel. ( ) xj X ij h j

16 The main idea The local rodeo algorithm for the kernel density estim The rodeo statistics for the kernel density estimator The density estimation Rodeo is based on the statistics Z j = ˆf H (x) h j ( ) xj X = 1 n K ij h j d ( ) xk X ( ) ik K 1 n xj X i=1 K ij h k n h j k=1 For the conditional variance term s 2 j = Var(Z j X 1,..., X n ) ( ) 1 n = Var Z ji X 1,..., X n = 1 n n Var(Z j1 X 1,..., X n ) i=1 Here, we used the sample variance of the Z ji to estimate Var(Z j1 ). n i=1 Z ji

17 Density rodeo algorithms The main idea The local rodeo algorithm for the kernel density estim Density Rodeo: Hard thresholding version 1. Select parameter 0 < β < 1 and initial bandwidth h 0, where h 0 = c 0 /log log n for some constant c 0. Let c n be a sequence, s.t. c n = O(1). 2. Initialize the bandwidths, and activate all dimensions: (a) h j = h 0, j = 1, 2,..., d. (b) A = {1, 2,..., d}. 3. While A is nonempty, do for each j A (a) Compute the derivative and variance: Z j and s j. (b) Compute the threshold λ j = s j 2 log(ncn ). (c) If Z j > λ j, then set h j βh j ; Otherwise remove j from A. 4. Output bandwidths h and the estimator f(x) = ˆf H (x)

18 The purpose of the analysis The analysis is trying to characterize the asymptotic aspects of the selected bandwidths the asymptotic running time (efficiency) the convergence rate of the risk (accuracy) To make the theoretical results more realistic, a key aspect of our analysis is that we allow the dimension d to increase with the sample size n! For this, we need to make some assumptions about the unknown density function to take the increasing dimension into account.

19 Assumptions Background (A1) Kernel assumption: uu T K(u)du = v 2 I d and v 2 < and K 2 (u)du = R(K) < (A2) Dimension assumption: d log d = O(log n) (A3) Initial bandwidth assumption: h (0) j = c 0 /log log n for (j = 1,..., d) Combing with A2, this implies that lim n n d j=1 h(0) j = (A4) Sparsity assumption: f(x) g(x 1,..., x r )h(x) where h jj (x) = 0, and satisfies r = O(1) (A5) Hessian assumption: tr(h T R (u)h R (u))du <, and f jj (x) 0 for j = 1, 2,..., r where H R (u) represents the Hessian for the relevant dimensions

20 Derivatives of both relevant and irrelevant dimensions Key Lemma: Under assumptions A1 A5, suppose that x is interior to the support of f. Suppose that K is a product kernel with bandwidth matrix H (s) = diag(h (s) 1,..., h(s) d ). Then µ (s) j = h (s) j For j R we have µ (s) j = h (s) j E[ ˆf H (s)(x) f(x)] = o P (h (s) j ) for all j R c E[ ˆf H (s)(x) f(x)] = h (s) j v 2 f jj (x) + o P (h (s) j ). Thus, for any integer s > 0, h s = h 0 β s, each j > r satisfies µ (s) j = o P (h (s) j ) = o P (h (0) j ).

21 Main theorem Background Main Theorem: Suppose that r = O(1) and (A1) (A5) hold. In addition, suppose that A min = min j r f jj (x) = Ω(1) and A max = max j r f jj (x) = Õ(1). Then, for every ɛ > 0, the number of iterations T n until the Rodeo stops satisfies ( 1 P 4 + r log 1/β(n 1 ɛ a n ) T n 1 ) 4 + r log 1/β(n 1+ɛ b n ) 1 where a n = Ω(1) and b n = Õ(1). More over, the algorithm outputs bandwidths h that satisfy ( ) P h j = h (0) j for all j > r 1 Also, we have ( ) P h (0) j (nb n ) 1/(4+r) ɛ h j h (0) j (na n ) 1/(4+r)+ɛ for all j r 1 assuming that h (0) j is defined as in A3.

22 Convergence rate of the risk Theorem 2 Under the same condition of the main theorem, the risk R h of the density Rodeo estimator satisfies for every ɛ > 0. R h = ÕP (n 4/(4+r)+ɛ) We write Y n = ÕP (a n ) to mean that Y n = O(b n a n ) where b n is logarithmic in n. As noted earlier, we write a n = Ω(b n ) if lim inf n a n bn > 0; similarly an = Ω(b n ) if a n = Ω(b n c n ) where c n is logarithmic in n.

23 Possible extensions Different versions of the density rodeo algorithm Using other distributions as irrelevant dimensions The density rodeo algorithm could be extended in many ways the soft-thresholding version the global version the reverse version the bootstrapping version the rodeo algorithm for local linear density estimator the greedier version using other distributions as irrelevant

24 Global density rodeo Different versions of the density rodeo algorithm Using other distributions as irrelevant dimensions The idea is by averaging the test statistics for multiple evaluation points x 1,..., x k sampled from the empirical distribution. To avoid the cancellation problem, the statistic is squared, let Z j (x i ) represents the derivative for the i-th evaluation point with respect to the bandwidth h j. We define the test statistic while s j = T j = 1 m m Zj 2 (x k ), k=1 j = 1,..., d Var(T j ) = 1 2tr(Cj 2 m ) + 4 ˆµ j T C j ˆµ j where ˆµ = 1 m m i=1 Z j(x i ). The threshold is λ j = s 2 j + 2s j log(ncn )

25 The bootstrapping version Different versions of the density rodeo algorithm Using other distributions as irrelevant dimensions Instead of derive the expression explicitly, bootstrapping can be used to evaluate the variance of Z j. Bootstrapping Method to calculate the s 2 j 1. Draw a sample X 1,..., X n of size n, with replacement: Loop for i = 1,..., B, Compute the estimate Z ji from data X 1,..., X n 2. Compute the bootstrapped variance s 2 j = 1 B B b=1 (Z ji Z j ) 2. where Zj = 1 B B b=1 Ẑ j 3. Output the resulted s 2 j.

26 The reverse and greedier version Different versions of the density rodeo algorithm Using other distributions as irrelevant dimensions Reverse density Rodeo: Instead of using a sequence of decreasing bandwidths. On the contrary, we could begin from a very small bandwidth, and use a sequence of increasing bandwidths to estimate the optimal value Greedier density rodeo: Instead of decaying all the bandwidths, only the bandwidths associated the largest Z j /λ j quantities is decayed Hybrid density Rodeo: Different variations could be combined arbitrarily

27 Different versions of the density rodeo algorithm Using other distributions as irrelevant dimensions Using other distributions as irrelevant dimensions We can use a general parametric distribution h(x) as irrelevant dimensions. The key trick is that a new semi-parametric density estimator will be used ˆf H (x) = ĥ(x) n i=1 K H(X i x) n K H (u x)ĥ(u)du where ĥ(x) is a parametric density estimator at point x. The motivation of this estimator comes from local likelihood method (Loader 1996). We see that, for one dimensional case, starting from a large bandwidth, if the true function is h(x), the algorithm will tend to freeze the bandwidth decaying process immediately.

28 Experimental design The density rodeo algorithms were applied on both synthetic and real data, including one-dimensional, two-dimensional, high-dimensional and very high-dimensional examples to measure the distance between the estimated density function with the true density function. The Hellinger distance are used ( D( ˆf f) = ˆf(x) 2 ˆf(x) f(x)) dx = 2 2 f(x) f(x) dx Assuming that we have altogether m evaluation points, then the hellinger distance could be numerically calculated by the Monte Carlo integration D( ˆf f) 2 2 m ˆfH (X i ) m f(x i ) i=1

29 One-dimensional example: strongly skewed density Strongly skewed density: We simulated 200 samples from the skewed distribution. The boxplot of the Hellinger distance is produced based on 100 simulations. Strongly skewed This density is chosen to resemble to lognormal distribution, it distributes as ( 7 ( 1 X 8 N 3 ( 2 ) ( ) ) 2i 2 3 )i 1, 3 i=0 The true density is plotted density x

30 Strongly skewed density: result Unbiased Cross Validation Local kernel density Rodeo density density True density Unbiased C.V x, bandwidth = Global kernel density Rodeo True density Global Rodeo density bandwidth True density Local Rodeo x Bandwidth estimated Unibased CV Local Rodeo Global Rodeo x, bandwidth = 0.05 x

31 Two-dimensional example: Combined Beta and Uniform Combined Beta distribution with unifrom distribution as irrelevant dimension. We simulate a 2-dimensional dataset with n = 500 points. true relevant dimension The two dimensions are independently generated as X Beta(1, 2) + 1 Beta(10, 10) 3 X 2 Uniform(0, 1) The true density for the relevant dimension is density x, bandwidth =

32 Density Density Background Combined Beta and Uniform: result The first plot is the rodeo result, the second plot is the result fitted by the built-in function KDE2d (MASS package in R) relevant dimension irrelevant dimension relevant dimension irrelevant dimension

33 Combined Beta and Uniform: marginal distribution True relevant density True irrelevant dimension density density x x relevant by Rodeo irrelvant by Rodeo density relevant dimension density irrelevant dimension Numerically integrated marginal distributions based on the perspective plots of the two estimators (not normalized). relevant by KDE2d irrelevant by KDE2d density density relevant dimension irrelevant dimension

34 Two-dimensional example: geyser data Geyser Data: A version of the eruptions data from the Old Faithful geyser in Yellowstone National Park, Wyoming. ( Azzalini and Bowman 1990) and is of continuous measurement from August 1 to August 15, There are altogether n = 299 samples and two variables. Duration = the numeric eruption time in minutes. waiting = the waiting time to next eruption.

35 Density first dimension Density first dimension Background Geyser data: result The first plot is the rodeo result, the second plot is the result fitted by the built-in function KDE2d (MASS package in R) second dimension second dimension

36 Geyser data: contour plot The first plot is the contour plot fitted by the built-in function KDE2d (MASS package in R), the second one is fitted by the rodeo algorithm second dimension second dimension first dimension first dimension

37 High dimensional example 30-dimensional example: We generate 30-dimensional synthetic dataset with r = 5 relevant dimensions (n = 100, with 30 trials). The relevant dimensions are generated as X i N (0.5, (0.02i) 2 ), for i = 1,..., 5 while the irrelevant dimensions are generated as X i Uniform(0, 1), for i = 6,..., 30 The test point is x = ( 1 2,..., 1 2 ).

38 30-dimensional example: result The Rodeo path for the 30-dimensional synthetic dataset and the boxplot of the selected bandwidths for 30 trials Bandwidth selected bandwidths Rodeo Step X1 X4 X7 X10 X14 X18 X22 X26 X30 dimensions

39 Very high dimensional example: image processing The algorithm was run on 2200 grayscale images of 1s and 2s, each with 256 = pixels with some unknown background noise; thus this is a 256-dimensional density estimation problem. A test point and the output bandwidth plot are shown here

40 Image processing example: evolution plot The output bandwidth plots sampled at the Rodeo step 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. Which visualizes the evolution of the bandwidths and could be viewed as a dynamic process for feature selection the earlier a dimension s bandwidth decays, the more informative it is

41 A example using Gaussian as irrelevant Using Gaussian as irrelevant dimensions: We apply the semiparametric rodeo algorithm on both 15-dimensional and 20-dimensional synthetic datasets with r = 5 relevant dimensions (n = 1000). When using gaussian distributions as irrelevant dimensions, the relevant dimensions are generated as X i Uniform(0, 1), for i = 1,..., 5 while the irrelevant dimensions are generated as X i N (0.5, (0.05i) 2 ), for i = 6,..., d The test point is x = ( 1 2,..., 1 2 ).

42 Using Gaussian as irrelevant: result The Rodeo path for the 15-dimensional synthetic data(left) and for the 20-dimensional data (Right) Bandwidth Bandwidth Rodeo Step Rodeo Step

43 Using Gaussian as irrelevant: one-dimensional case (I) True density fitted by Rodeo density density bandwidth x Bandwidth estimated Density x By unbiased CV 1000 one-dimensional data points with x i Uniform(0, 1) x N = 1000 Bandwidth =

44 Using Gaussian as irrelevant: one-dimensional case (II) Gaussian fitted by Rodeo density bandwidth density x Bandwidth estimated C x correction factor 1000 one-dimensional data points with x i N (0, 1) x test

45 Summary Background This work adapts the general rodeo framework to solve density estimation problems The sparsity assumption is crucial to guarantee the success of the density rodeo algorithm The density rodeo algorithm is efficient on high-dimensional problems both theoretically and empirically. The rodeo framework can utilize current available density estimators, the implementation is simple Future work: develop the rodeo algorithms for the other kinds of problems, e.g. classification.