... SPARROW. SPARse approximation Weighted regression. Pardis Noorzad. Department of Computer Engineering and IT Amirkabir University of Technology

Transcription

1 ..... SPARROW SPARse approximation Weighted regression Pardis Noorzad Department of Computer Engineering and IT Amirkabir University of Technology Université de Montréal March 12, 2012 SPARROW 1/47

2 ..... Outline Introduction Motivation Local Methods SPARROW is a Local Method Defining the Effective Weights Defining the Observation Weights SPARROW 2/47

3 .. Motivation Problem setting... Given D := {(x i, y i ) : i = 1,, N} yi R is the output at the input xi := [x i1,, x im ] T R M our task is to estimate the regression function f : R M R such that y i = f(x i ) + ϵ i the ϵ i s are independent with zero mean. SPARROW 5/47

4 .. Motivation Global methods... In parametric approaches, the regression function is known for e.g., in multiple linear regression (MLR) we assume M f(x 0 ) = β j x 0j + ϵ j=1 we can also add higher order terms but still have a model that is linear in the parameters β j, γ j f(x 0 ) = M j=1 ( βj x 0j + γ j x 2 ) 0j + ϵ SPARROW 6/47

5 .. Motivation Global methods Continued... Example of a global nonparametric approach: ϵ-support vector regression (ϵ-svr) (Smola and Schölkopf, 2004) f(x 0 ) = N β j K(x 0, x i ) + ϵ i=1 SPARROW 7/47

6 .. Local Methods Local methods... A successful nonparametric approach to regression: local estimation (Hastie and Loader, 1993; Härdle and Linton, 1994; Ruppert and Wand, 1994) In local methods: f(x 0 ) = N l i (x 0 )y i + ϵ i=1 SPARROW 9/47

7 .. Local Methods Local methods Continued... For e.g. in k-nearest neighbor regression (k-nnr) f(x 0 ) = N i=1 α i (x 0 ) N p=1 α p(x 0 ) y i where α i (x 0 ) := I Nk (x 0)(x i ) Nk (x 0 ) D is the set of the k-nearest neighbors of x 0 SPARROW 10/47

8 .. Local Methods Local methods Continued... In weighted k-nnr (Wk-NNR), f(x 0 ) = N i=1 α i (x 0 ) N p=1 α p(x 0 ) y i α i (x 0 ) := S(x 0, x i ) 1 I Nk (x 0 )(x i ) S(x0, x i ) = (x 0 x i ) T V 1 (x 0 x i ) is the scaled Euclidean distance SPARROW 11/47

9 .. Local Methods Local methods Continued... Just so you know, here s another example of a local method: additive model (AM) (Buja et al., 1989) f(x 0 ) = M f j (x 0j ) + ϵ j=1 Estimate univariate functions of predictors locally SPARROW 12/47

10 .. Local Methods Local methods Continued... In local methods: estimate the regression function locally by a simple parametric model In local polynomial regression: estimate the regression function locally, by a Taylor polynomial This is what happens in SPARROW, as we will explain SPARROW 13/47

11 .. SPARROW is a Local Method... SPARROW 16/47

12 .. SPARROW is a Local Method I meant this sparrow... SPARROW 17/47

13 .. SPARROW is a Local Method... SPARROW is a local method Before we get into the details, see a few examples showing benefits of local methods then we ll talk about SPARROW SPARROW 18/47

14 Goal function Dataset Figure: Our generated dataset. y i = f(x i ) + ϵ i, where f(x) = (x 3 + x 2 ) I(x) + sin(x) I( x).

15 Dataset Goal function 0.4 MLR:1st MLR:2nd MLR:3rd Figure: Multiple linear regression with first-, second-, and third-order terms.

16 Dataset Goal function ε SVR Figure: ϵ-support vector regression with an RBF kernel.

17 Dataset Goal function 4 NNR Figure: 4-nearest neighbor regression.

18 .. Defining the Effective Weights... Effective weights in SPARROW In local methods: f(x 0 ) = N l i (x 0 )y i + ϵ i=1 Now we define l i (x 0 ) SPARROW 24/47

19 .. Defining the Effective Weights... Local estimation by a Taylor polynomial To locally estimate the regression function near x 0 let us approximate f(x) by a second-degree Taylor polynomial about x 0 P 2 (x) = ϕ + (x x 0 ) T θ (x x 0) T H(x x 0 ) (1) ϕ := f(x 0 ), θ := f(x 0 ) is the gradient of f(x), H := 2 f(x 0 ) is its Hessian both evaluated at x 0 SPARROW 25/47

20 .. Defining the Effective Weights... Local estimation by a Taylor polynomial Continued P 2 (x) = ϕ + (x x 0 ) T θ (x x 0) T H(x x 0 ) We need to solve the locally weighted least squares problem { min α i yi P 2 (x i ) } 2 (2) ϕ,θ,h i Ω SPARROW 26/47

21 .. Defining the Effective Weights... Local estimation by a Taylor polynomial Continued Express (2) as min A 1/2{ y XΘ(x 0 ) } 2 Θ(x 0 ) (3) a ii = α i, y := [y 1, y 2,, y N ] T 1 (x 1 x 0 ) T vech T {(x 1 x 0 )(x 1 x 0 ) T } X :=... 1 (x N x 0 ) T vech T {(x N x 0 )(x N x 0 ) T } parameter supervector: Θ(x 0 ) := [ ϕ, θ, vech(h) ] T SPARROW 27/47

22 .. Defining the Effective Weights... Local estimation by a Taylor polynomial Continued The solution: Θ(x 0 ) = ( X T AX ) 1 X T Ay And so the local quadratic estimate is ˆϕ = ˆf(x 0 ) = e T 1 ( X T AX ) 1 X T Ay Since f(x 0 ) = N i=1 l i(x 0 )y i, the vector of effective weights for SPARROW is [l 1 (x 0 ),, l N (x 0 )] T = A T X ( X T AX ) 1 e1 SPARROW 28/47

23 .. Defining the Effective Weights... Local estimation by a Taylor polynomial Continued The local constant regression estimate is ˆf(x 0 ) = (1 T A1) 1 1 T Ay = N i=1 α i (x 0 ) N k=1 α k(x 0 ) y i. Look familiar? SPARROW 29/47

24 .. Defining the Observation Weights... Observation weights in SPARROW We have to assign the weights here { min α i yi f(x i ) } 2 ϕ,θ,h i Ω that is, the diagonal elements of A min A 1/2{ y XΘ(x 0 ) } 2 Θ(x 0 ) (4) SPARROW 31/47

25 .. Defining the Observation Weights... Observation weights in SPARROW Continued To find α i we solve the following problem (Chen et al., 1995) min α 1 subject to x 0 Dα 2 α R N 2 σ (5) σ > 0 limits [ the maximum approximation ] error and D := x1 x 1, x 2 x 2,, x N x N SPARROW 32/47

26 .. Defining the Observation Weights Power family of penalties l p norms raised to the pth power... ( ) x p p = x i p i (6) For 1 p <, (6) is convex. 0 < p 1, is the range of p useful for measuring sparsity. SPARROW 33/47

27 Figure: As p goes to 0, x p becomes the indicator function and x p becomes a count of the nonzeros in x (Bruckstein et al., 2009).

28 .. Defining the Observation Weights... Representation by sparse approximation Continued To motivate this idea let s look at feature learning with sparse coding, and sparse representation classification (SRC) an example of exemplar-based sparse approximation SPARROW 35/47

29 .. Defining the Observation Weights... Unsupervised feature learning Application to image classification x 0 = Dα An example is the recent work by Coates and Ng (2011). where x 0 is the input vector could be a vectorized image patch, or a SIFT descriptor α is the higher-dimensional sparse representation of x0 D is usually learned SPARROW 36/47

30 Figure: Image classification (Coates et al., 2011).

31 .. Defining the Observation Weights Multiclass classification (Wright et al., 2009)... D := { (x i, y i ) : x i R m, y i {1,, c}, i {1,, N} } Given a test sample x 0 1. Solve min α R N α 1 subject to x 0 Dα 2 2 σ 2. Define {α y : y {1,, c}} where [α y ] i = α i if x i belongs to class y, o.w Construct X (α) := {ˆx y (α) = Dα y, y {1,, c} } 4. Predict ŷ := arg min y {1,...,c} x 0 ˆx y (α) 2 2 SPARROW 38/47

32 Figure: SRC on handwritten image dataset.

33 Figure: SVM with linear kernel on handwritten image dataset.

34 ..... Back to SPARROW with evaluation on the MPG dataset Auto MPG Data Set from the UCI Machine Learning Repository (Frank and Asuncion, 2010) The data concerns city-cycle fuel consumption in miles per gallon, to be predicted in terms of 3 multivalued discrete and 4 continuous attributes. number of instances: 392 number of attributes: 7 (cylinders, displacement, horsepower, weight, acceleration, model year, origin) SPARROW 42/47

35 MSE NWR 4 NNR W4 NNR SPARROW MLR LASSO Figure: Average mean squared error values achieved by various methods over 10-fold cross-validation.

36 ..... Looking ahead What is causing the success of SPARROW and SRC? How important is the bandwidth? What about in SRC? SPARROW 44/47

37 References References I Alfred M. Bruckstein, David L. Donoho, and Michael Elad. From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Review, 51(1):34 81, Andreas Buja, Trevor Hastie, and Robert Tibshirani. Linear smoothers and additive models. The Annals of Statistics, 17(2): , Scott S. Chen, David L. Donoho, and Michael A. Saunders. Atomic decomposition by basis pursuit. Technical Report 479, Department of Statistics, Stanford University, May Adam Coates and Andrew Ng. The importance of encoding versus training with sparse coding and vector quantization. In International Conference on Machine Learning (ICML), pages , Adam Coates, Honglak Lee, and Andrew Y. Ng. Self-taught learning: Transfer learning from unlabeled data. In International Conference on AI and Statistics, SPARROW 45/47

38 References References II A. Frank and A. Asuncion. UCI machine learning repository, URL Wolfgang Härdle and Oliver Linton. Applied nonparametric methods. Technical Report 1069, Yale University, T. J. Hastie and C. Loader. Local regression: Automatic kernel carpentry. Statistical Science, 8(2): , D. Ruppert and M. P. Wand. Multivariate locally weighted least squares regression. The Annals of Statistics, 22: , Alex J. Smola and Bernhard Schölkopf. A tutorial on support vector regression. Statistics and Computing, 14: , August John Wright, Allen Y. Yang, Arvind Ganesh, S. Shankar Sastry, and Yi Ma. Robust face recognition via sparse representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 31: , SPARROW 46/47

39 References Acknowledgements This is ongoing work carried out under the supervision of Prof. Bob L. Sturm of Aalborg University Copenhagen. Thanks to Isaac Nickaein and Sheida Bijani for helping out with the slides. SPARROW 47/47