Supervised Dimension Reduction:

Transcription

1 Supervised Dimension Reduction: A Tale of Two Manifolds S. Mukherjee, K. Mao, F. Liang, Q. Wu, M. Maggioni, D-X. Zhou Department of Statistical Science Institute for Genome Sciences & Policy Department of Computer Science Department of Mathematics Duke University November 4, 2009

2 Supervised dimension reduction Information and sufficiency A fundamental idea in statistical thought is to reduce data to relevant information. This was the paradigm of R.A. Fisher (beloved Bayesian) and goes back to at least Adcock 1878 and Edgeworth 1884.

3 Supervised dimension reduction Information and sufficiency A fundamental idea in statistical thought is to reduce data to relevant information. This was the paradigm of R.A. Fisher (beloved Bayesian) and goes back to at least Adcock 1878 and Edgeworth X 1,...,X n drawn iid form a Gaussian can be reduced to µ,σ 2.

4 Supervised dimension reduction Regression Assume the model with X X R p and Y R. Y = f (X) + ε, IEε = 0,

5 Supervised dimension reduction Regression Assume the model with X X R p and Y R. Y = f (X) + ε, IEε = 0, Data D = {(x i,y i )} n iid i=1 ρ(x,y ).

6 Supervised dimension reduction Dimension reduction If the data lives in a p-dimensional space X IR p replace X with Θ(X) IR d, p d.

7 Supervised dimension reduction Dimension reduction If the data lives in a p-dimensional space X IR p replace X with Θ(X) IR d, p d. My belief: physical, biological and social systems are inherently low dimensional and variation of interest in these systems can be captured by a low-dimensional submanifold.

8 Supervised dimension reduction Supervised dimension reduction (SDR) Given response variables Y 1,...,Y n IR and explanatory variables or covariates X 1,...,X n X R p Y i = f (X i ) + ε i, ε i iid No(0,σ 2 ).

9 Supervised dimension reduction Supervised dimension reduction (SDR) Given response variables Y 1,...,Y n IR and explanatory variables or covariates X 1,...,X n X R p Y i = f (X i ) + ε i, ε i iid No(0,σ 2 ). Is there a submanifold S S Y X such that Y X P S (X)?

10 Supervised dimension reduction Visualization of SDR 20 (a) Data 1 (b) Diffusion map z y x Dimension Dimension (c) GOP 1 (d) GDM Dimension Dimension Dimension Dimension 1

11 Supervised dimension reduction Linear projections capture nonlinear manifolds In this talk P S (X) = B T X where B = (b 1,...,b d ).

12 Supervised dimension reduction Linear projections capture nonlinear manifolds In this talk P S (X) = B T X where B = (b 1,...,b d ). Semiparametric model Y i = f (X i ) + ε i = g(b1 T X i,...,bd T X i) + ε i, span B is the dimension reduction (d.r.) subspace.

13 Learning gradients SDR model Semiparametric model Y i = f (X i ) + ε i = g(b T 1 X i,...,b T d X i) + ε i, span B is the dimension reduction (d.r.) subspace.

14 Learning gradients SDR model Semiparametric model Y i = f (X i ) + ε i = g(b T 1 X i,...,b T d X i) + ε i, span B is the dimension reduction (d.r.) subspace. Assume marginal distribution ρ X is concentrated on a manifold M IR p of dimension d p.

15 Learning gradients Gradients and outer products Given a smooth function f the gradient is ( ) f (x) = f (x) f (x) T. x 1,..., x p

16 Learning gradients Gradients and outer products Given a smooth function f the gradient is ( ) f (x) = f (x) f (x) T. x 1,..., x p Define the gradient outer product matrix Γ f Γ ij = (x) f (x)dρ x i x X (x), j X Γ = E[( f ) ( f )].

17 Learning gradients GOP captures the d.r. space Suppose y = f (X) + ε = g(b1 T X,...,bd T X) + ε.

18 Learning gradients GOP captures the d.r. space Suppose y = f (X) + ε = g(b1 T X,...,bd T X) + ε. Note that for B = (b 1,...,b d ) λ i b i = Γb i.

19 Learning gradients GOP captures the d.r. space Suppose y = f (X) + ε = g(b1 T X,...,bd T X) + ε. Note that for B = (b 1,...,b d ) λ i b i = Γb i. For i = 1,..,d f (x) v i = v T i ( f (x)) 0 b T i Γb i 0. If w b i for all i then w T Γw = 0.

20 Learning gradients Statistical interpretation Linear case y = β T x + ε, ε iid No(0,σ 2 ). Ω = cov (E[X Y ]), Σ X = cov (X), σ 2 = var (Y ). Y

21 Learning gradients Statistical interpretation Linear case y = β T x + ε, ε iid No(0,σ 2 ). Ω = cov (E[X Y ]), Σ X = cov (X), σ 2 Y Γ = σ 2 Y ( 1 σ2 σ 2 Y ) 2 Σ 1 X ΩΣ 1 X = var (Y ). σ 2 Σ 1ΩΣ 1. Y X X

22 Learning gradients Statistical interpretation For smooth f (x) y = f (x) + ε, ε iid No(0,σ 2 ). Ω = cov (E[X Y ]) not so clear.

23 Learning gradients Nonlinear case Partition into sections and compute local quantities X = I i=1 χ i

24 Learning gradients Nonlinear case Partition into sections and compute local quantities I X = i=1 χ i Ω i = cov (E[X χi Y χi ])

25 Learning gradients Nonlinear case Partition into sections and compute local quantities I X = i=1 χ i Ω i = cov (E[X χi Y χi ]) Σ i = cov (X χi )

26 Learning gradients Nonlinear case Partition into sections and compute local quantities I X = i=1 χ i Ω i = cov (E[X χi Y χi ]) Σ i = cov (X χi ) σi 2 = var (Y χi )

27 Learning gradients Nonlinear case Partition into sections and compute local quantities I X = i=1 χ i Ω i = cov (E[X χi Y χi ]) Σ i = cov (X χi ) σi 2 = var (Y χi ) m i = ρ X (χ i ).

28 Learning gradients Nonlinear case Partition into sections and compute local quantities I X = i=1 χ i Ω i = cov (E[X χi Y χi ]) Σ i = cov (X χi ) σi 2 = var (Y χi ) m i = ρ X (χ i ). Γ I i=1 m i σi 2 Σ 1 i Ω i Σ 1 i.

29 Learning gradients Estimating the gradient Taylor expansion y i f (x i ) f (x j ) + f (x j ),x j x i y j + f (x j ),x j x i if x i x j.

30 Learning gradients Estimating the gradient Taylor expansion y i f (x i ) f (x j ) + f (x j ),x j x i y j + f (x j ),x j x i if x i x j. Let f f the following should be small w ij (y i y j f (x j ),x j x i ) 2, i,j w ij = 1 s p+2 exp( x i x j 2 /2s 2 ) enforces x i x j.

31 Learning gradients Estimating the gradient The gradient estimate fd = arg min 1 n n 2 f H p i,j=1 w ij (y i y j ( ) 2 f (x j )) T (x j x i ) + λ f 2 K where f K is a smoothness penalty, reproducing kernel Hilbert space norm.

32 Learning gradients Estimating the gradient The gradient estimate fd = arg min 1 n n 2 f H p i,j=1 w ij (y i y j ( ) 2 f (x j )) T (x j x i ) + λ f 2 K where f K is a smoothness penalty, reproducing kernel Hilbert space norm.

33 Learning gradients Computational efficiency The computation requires fewer than n 2 parameters and is O(n 6 ) time and O(pn) memory fd (x) = n c i,d K(x i,x) i=1 c D = (c 1,D,...,c n,d ) T R np.

34 Learning gradients Computational efficiency The computation requires fewer than n 2 parameters and is O(n 6 ) time and O(pn) memory fd (x) = n c i,d K(x i,x) i=1 c D = (c 1,D,...,c n,d ) T R np. Define gram matrix K where K ij = K(x i,x j ) ˆΓ = c D Kc T D.

35 Learning gradients Estimates on manifolds Mrginal distribution ρ X is concentrated on a compact Riemannian manifold M IR d with isometric embedding ϕ : M R p and metric d M and dµ is the uniform measure on M. Assume regular distribution (i) The density ν(x) = dρ X (x) dµ exists and is Hölder continuous (c 1 > 0 and 0 < θ 1) ν(x) ν(u) c 1 dm θ (x, u) x, u M. (ii) The measure along the boundary is small: (c 2 > 0) ρ M ` x M : dm (x, M) t c 2 t t > 0.

36 Learning gradients Convergence to gradient on manifold Theorem Under above regularity conditions on ρ X and f C 2 (M), with probability 1 δ ( ) (dϕ) 1 ( ) fd M f 2 L C log n 1 2 d. ρ M δ where (dϕ) (projection onto tangent space) is the dual of the map dϕ.

37 Baysian Mixture of Inverses Principal components analysis (PCA) Algorithmic view of PCA: 1. Given X = (X 1,...,X n ) a p n matrix construct ˆΣ = (X X)(X X) T

38 Baysian Mixture of Inverses Principal components analysis (PCA) Algorithmic view of PCA: 1. Given X = (X 1,...,X n ) a p n matrix construct ˆΣ = (X X)(X X) T 2. Eigen-decomposition of ˆΣ λ i v i = ˆΣv i.

39 Baysian Mixture of Inverses Probabilistic PCA X IR p is charterized by a multivariate normal µ IR p A IR p d IR p p ν IR d. X No(µ + Aν, ), ν No(0,I d )

40 Baysian Mixture of Inverses Probabilistic PCA X IR p is charterized by a multivariate normal µ IR p A IR p d IR p p ν IR d. ν is a latent variable X No(µ + Aν, ), ν No(0,I d )

41 Baysian Mixture of Inverses SDR model Semiparametric model Y i = f (X i ) + ε i = g(b T 1 X i,...,b T d X i) + ε i, span B is the dimension reduction (d.r.) subspace.

42 Baysian Mixture of Inverses Principal fitted components (PFC) Define X y (X Y = y) and specify multivariate normal distribution µ IR p A IR p d ν y IR d. X y No(µ y, ), µ y = µ + Aν y

43 Baysian Mixture of Inverses Principal fitted components (PFC) Define X y (X Y = y) and specify multivariate normal distribution µ IR p A IR p d ν y IR d. B = 1 A. X y No(µ y, ), µ y = µ + Aν y

44 Baysian Mixture of Inverses Principal fitted components (PFC) Define X y (X Y = y) and specify multivariate normal distribution µ IR p A IR p d ν y IR d. B = 1 A. X y No(µ y, ), µ y = µ + Aν y Captures global linear predictive structure. Does not generalize to manifolds.

45 Baysian Mixture of Inverses Mixture models and localization A driving idea in manifold learning is that manifolds are locally Euclidean.

46 Baysian Mixture of Inverses Mixture models and localization A driving idea in manifold learning is that manifolds are locally Euclidean. A driving idea in probabilistic modeling is that mixture models are flexible and can capture nonparametric distributions.

47 Baysian Mixture of Inverses Mixture models and localization A driving idea in manifold learning is that manifolds are locally Euclidean. A driving idea in probabilistic modeling is that mixture models are flexible and can capture nonparametric distributions. Mixture models can capture local nonlinear predictive manifold structure.

48 Baysian Mixture of Inverses Model specification X y No(µ yx, ) µ yx = µ + Aν yx ν yx G y G y : density indexed by y having multiple clusters µ IR p ε N(0, ) with IR p p A IR p d ν xy IR d.

49 Baysian Mixture of Inverses Dimension reduction space Proposition For this model the d.r. space is the span of B = 1 A Y X d = Y ( 1 A) T X.

50 Baysian Mixture of Inverses Sampling distribution Define ν i ν yi x i. Sampling distribution for data x i (y i,µ,ν i,a, ) N(µ + Aν i, ) ν i G yi.

51 Baysian Mixture of Inverses Categorical response: modeling G y Y = {1,...,C}, so each category has a distribution ν i (y i = k) G k, c = 1,...,C.

52 Baysian Mixture of Inverses Categorical response: modeling G y Y = {1,...,C}, so each category has a distribution ν i (y i = k) G k, c = 1,...,C. ν i modeled as a mixture of C distributions G 1,...,G C with a Dirichlet process model for ech distribution G c DP(α 0,G 0 ).

53 Baysian Mixture of Inverses Categorical response: modeling G y Y = {1,...,C}, so each category has a distribution ν i (y i = k) G k, c = 1,...,C. ν i modeled as a mixture of C distributions G 1,...,G C with a Dirichlet process model for ech distribution G c DP(α 0,G 0 ). Goto board.

54 Baysian Mixture of Inverses Likelihood Lik(data θ) Lik(data A,,ν 1,...,ν n,µ) Lik(data θ) det( 1 ) n 2 [ ] exp 1 n (x i µ Aν i ) T 1 (x i µ Aν i ). 2 i=1

55 Baysian Mixture of Inverses Posterior inference Given data P θ Post(θ data) Lik(θ data) π(θ).

56 Baysian Mixture of Inverses Posterior inference Given data P θ Post(θ data) Lik(θ data) π(θ). 1. P θ provides estimate of (un)certainty on θ

57 Baysian Mixture of Inverses Posterior inference Given data P θ Post(θ data) Lik(θ data) π(θ). 1. P θ provides estimate of (un)certainty on θ 2. Requires prior on θ

58 Baysian Mixture of Inverses Posterior inference Given data P θ Post(θ data) Lik(θ data) π(θ). 1. P θ provides estimate of (un)certainty on θ 2. Requires prior on θ 3. Sample from P θ?

59 Baysian Mixture of Inverses Markov chain Monte Carlo No closed form for P θ.

60 Baysian Mixture of Inverses Markov chain Monte Carlo No closed form for P θ. 1. Specify Markov transition kernel K(θ t,θ t+1 ) with stationary distribution P θ.

61 Baysian Mixture of Inverses Markov chain Monte Carlo No closed form for P θ. 1. Specify Markov transition kernel K(θ t,θ t+1 ) with stationary distribution P θ. 2. Run the Markov chain to obtain θ 1,...,θ T.

62 Baysian Mixture of Inverses Sampling from the posterior Inference consists of drawing samples θ (t) = (µ (t),a (t), 1 (t),ν (t)) from the posterior.

63 Baysian Mixture of Inverses Sampling from the posterior Inference consists of drawing samples θ (t) = (µ (t),a (t), 1 (t),ν (t)) from the posterior. Define θ /µ (t) (A (t), 1 (t),ν (t)) θ /A (t) (µ (t), 1 (t),ν (t)) θ / 1 (t) (µ (t),a (t),ν (t) ) θ /ν (t) (µ (t),a (t), 1 (t) ).

64 Baysian Mixture of Inverses Gibbs sampling Conditional probabilities can be used to sample µ, 1,A ( ) ( ) µ (t+1) data,θ /µ (t) No data,θ /µ (t),

65 Baysian Mixture of Inverses Gibbs sampling Conditional probabilities can be used to sample µ, 1,A ( ) ( ) µ (t+1) data,θ /µ (t) No data,θ /µ (t), ) ) 1 (t+1) (data,θ / 1 (t) InvWishart (data,θ / 1 (t)

66 Baysian Mixture of Inverses Gibbs sampling Conditional probabilities can be used to sample µ, 1,A ( ) ( ) µ (t+1) data,θ /µ (t) No data,θ /µ (t), ) ) 1 (t+1) (data,θ / 1 (t) InvWishart (data,θ / 1 (t) ( ) ( ) A (t+1) data,θ /A (t) No data.θ /A (t).

67 Baysian Mixture of Inverses Gibbs sampling Conditional probabilities can be used to sample µ, 1,A ( ) ( ) µ (t+1) data,θ /µ (t) No data,θ /µ (t), ) ) 1 (t+1) (data,θ / 1 (t) InvWishart (data,θ / 1 (t) ( ) ( ) A (t+1) data,θ /A (t) No data.θ /A (t). Sampling ν (t) is more involved.

68 Baysian Mixture of Inverses Posterior draws from the Grassmann manifold Given samples ( 1 (t),a (t)) m t=1 compute B (t) = 1 (t) A (t).

69 Baysian Mixture of Inverses Posterior draws from the Grassmann manifold Given samples ( 1 (t),a (t)) m t=1 compute B (t) = 1 (t) A (t). Each B (t) is a subspace which is a point in the Grassmann manifold G (d,p). There is a Riemannian metric on this manifold. This has two implications.

70 Baysian Mixture of Inverses Posterior mean and variance Given draws (B (t) ) m t=1 the posterior mean and variance should be computed with respect to the Riemannian metric.

71 Baysian Mixture of Inverses Posterior mean and variance Given draws (B (t) ) m t=1 the posterior mean and variance should be computed with respect to the Riemannian metric. Given two subspaces W and U spanned by orthonormal bases W and V the Karcher mean is (I X(X T X) 1 X T )Y (X T Y ) 1 = UΣV T Θ = atan(σ) dist(w, V) = Tr(Θ 2 ).

72 Baysian Mixture of Inverses Posterior mean and variance The posterior mean subspace B Bayes = arg min B G (d,p) m dist(b i, B). i=1

73 Baysian Mixture of Inverses Posterior mean and variance The posterior mean subspace B Bayes = arg min B G (d,p) m dist(b i, B). i=1 Uncertainty var({b 1,, B m }) = 1 m m dist(b i, B Bayes ). i=1

74 Baysian Mixture of Inverses Distribution theory on Grassmann manifolds If B is a linear space of d central normal vectors in IR p with covariance matrix Σ the density of Grassmannian distribution G Σ w.r.t. reference measure G I is ( dg Σ det(x T ) d/2 X) ( X ) = dg I det(x T Σ 1, X) where X span(x) where X = (x 1,...x d ).

75 Results on data Swiss roll Swiss roll X 1 = t cos(t), X 2 = h, X 3 = t sin(t), X 4,...,10 iid No(0,1) where t = 3π 2 (1 + 2θ), θ Unif(0,1), h Unif(0,1) and Y = sin(5πθ) + h 2 + ε, ε No(0,0.01).

76 Results on data Swiss roll Pictures

77 Results on data Swiss roll Metric Projection of the estimated d.r. space ˆB = (ˆb 1,, ˆb d ) onto B 1 d d P Bˆbi 2 = 1 d i=1 d (BB T )ˆb i 2 i=1

78 Results on data Swiss roll Comparison of algorithms 1 Accuracy BMI BAGL SIR LSIR PHD SAVE Sample size

79 Results on data Swiss roll Posterior variance Boxplot for the Distances

80 Results on data Swiss roll Error as a function of d error 0.4 error number of e.d.r. directions number of e.d.r. directions

81 Results on data Digits Digits

82 Results on data Digits Two classification problems 3 vs. 8 and 5 vs. 8.

83 Results on data Digits Two classification problems 3 vs. 8 and 5 vs training samples from each class.

84 Results on data Digits BMI x x

85 Results on data Digits All ten digits digit Nonlinear Linear (± 0.01) 0.05 (± 0.01) (± 0.003) 0.03 (± 0.01) (± 0.02) 0.19 (± 0.02) (± 0.01) 0.17 (± 0.03) (± 0.02) 0.13 (± 0.03) (± 0.02) 0.21 (± 0.03) (± 0.01) (± 0.02) (± 0.01) 0.14 (± 0.02) (± 0.02) 0.20 (± 0.03) (± 0.02) 0.15 (± 0.02) average Table: Average classification error rate and standard deviation on the digits data.

86 Results on data Cancer Cancer classification n = 38 samples with expression levels for p = 7129 genes or ests 19 samples are Acute Myeloid Leukemia (AML) 19 are Acute Lymphoblastic Leukemia, these fall into two subclusters B-cell and T-cell.

87 Results on data Cancer Substructure captured 5 x x 10 4

88 The end Funding IGSP Center for Systems Biology at Duke NSF DMS