LECTURE : MANIFOLD LEARNING
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1 LECTURE : MANIFOLD LEARNING Rta Osadchy Some sldes are due to L.Saul, V. C. Raykar, N. Verma
2 Topcs PCA MDS IsoMap LLE EgenMaps Done!
3 Dmensonalty Reducton Data representaton Inputs are real-valued vectors n a hgh dmensonal space. Lnear structure Does the data lve n a low dmensonal subspace? Nonlnear structure Does the data lve on a low dmensonal submanfold?
4 Notatons Inputs (hgh dmensonal) x 1,x 2,,x n ponts n R D Outputs (low dmensonal) y 1,y 2,,y n ponts n R d (d<<d) Goals Nearby ponts reman nearby. Dstant ponts reman dstant.
5 Non-metrc MDS for manfolds? Rank orderng of Eucldean dstances s NOT preserved n manfold learnng.
6 Nonlnear Manfolds A PCA and MDS measure the Eucldean dstance What s mportant s the geodesc dstance Unroll the manfold
7 To preserve structure preserve the geodesc dstance and not the eucldean dstance.
8 Graph-Based Methods Tenenbaum et.al s Isomap Algorthm Global approach. Preserves global parwse dstances. Rowes and Saul s Locally Lnear Embeddng Algorthm Local approach Nearby ponts should map nearby Belkn and Nyog Laplacan Egenmaps Algorthm Local approach mnmzes approxmately the same value as LLE
9 Isomap - Key Idea: Use geodesc nstead of Eucldean dstances n MDS. For neghborng ponts Eucldean dstance s a good approxmaton to the geodesc dstance. For dstant ponts estmate the dstance by a seres of short hops between neghborng ponts. Fnd shortest paths n a graph wth edges connectng neghborng data ponts.
10 Step 1. Buld adjacency graph. Adjacency graph Vertces represent nputs. Undrected edges connect neghbours. Neghbourhood selecton Many optons: k-nearest neghbours, nputs wthn radus r, pror knowledge. Graph s dscretzed approxmaton of submanfold.
11 Buldng the graph Computaton knn scales navely as Faster methods explot data structures. Assumptons 1. Graph s connected. O( n 2 D) 2. Neghbourhoods on graph reflect neghbourhoods on manfold.
12 Step 2. Estmate geodescs Dynamc programmng Weght edges by local dstances. Compute shortest paths through graph. Geodesc dstances Estmate by lengths of shortest paths: denser samplng = better estmates. Computaton Djkstra s algorthm for shortest paths O(n 2 log n + n 2 k).
13 Step 3. Metrc MDS Embeddng Top d egenvectors of Gram matrx yeld embeddng. Dmensonalty Number of sgnfcant egenvalues yeld estmate of dmensonalty. Computaton Top d egenvectors can be computed n O(n 2 d).
14 Summary Algorthm 1. k nearest neghbours 2. shortest paths through graph 3. MDS on geodesc dstances
15 Swss Roll n (ponts) =1024 k (neghbors) =12
16 Isomap: Two-dmensonal embeddng of hand mages (from Josh. Tenenbaum, Vn de Slva, John Langford 2000) n =2000, k =6, D=64x64
17 Isomap: two-dmensonal embeddng of hand-wrtten 2 (from Josh. Tenenbaum, Vn de Slva, John Langford 2000) n =1000, r=4.2, D=20x20
18 Isomap: three-dmensonal embeddng of faces (from Josh. Tenenbaum, Vn de Slva, John Langford 2000) n =698, k=6
19 Propertes of Isomap Strengths : Preserves the global data structure Performs global optmzaton Non-parametrc (Only heurstc s neghbourhood sze) Weaknesses : Senstve to shortcuts Very slow
20 Spectral Methods Common framework 1. Derve sparse graph from knn. 2. Derve matrx from graph weghts. 3. Derve embeddng from egenvectors. Vared solutons Algorthms dffer n step 2. Types of optmzaton: shortest paths, least squares fts, semdefnte programmng.
21 Locally Lnear Embeddng (LLE) Assume that data les on a manfold: each sample and ts neghbors le on approxmately lnear subspace Idea: 1. Approxmate data by a set of lnear patches 2. Glue these patches together on a low dmensonal subspace s.t. neghborhood relatonshps between patches are preserved. Algorthm:
22 LLE at glance Steps 1. Nearest neghbour search. 2. Least squares fts. 3. Sparse egenvalue problem. Propertes Obtans hghly nonlnear embeddngs. Not prone to local mnma. Sparse graphs yeld sparse problems.
23 Step 1. Nearest neghbours search Effect of Neghbourhood Sze
24 Step 2. Compute weghts Characterze local geometry of each neghbourhood by weghts Wj. Compute weghts by reconstructng each nput (lnearly) from neghbours.
25 Lnear reconstructons Local lnearty Assume neghbours le on locally lnear patches of a low dmensonal manfold. Mnmze reconstructon error Each pont can be wrtten as a lnear combnaton of ts neghbors. The weghts chosen to mnmze the reconstructon error: mn x Wj x W j j 2
26 Least squares fts (Computng W j ) Local reconstructons Choose weghts to mnmze: Constrants W = 0 x j Set f s not a neghbor of j Weghts must sum to one: Wj = 1 j Local nvarance W j nvarance to translaton Φ( W ) = x W Optmal weghts are nvarant to rotaton, translaton, and scalng. x j j x j 2
27 Step 3. Fndng the Embeddng Low dmensonal representaton Map nputs to outputs: Mnmze reconstructon errors Optmze outputs for fxed weghts: x R Ψ ( y) = y W j D j y j 2 y R d Constrants: Center outputs on orgn y = 0 1 N Impose unt covarance matrx y y = I d
28 Mnmzaton Quadratc form: ( y y ) Ψ( y) = M M δ j j j j =δ W W + W W, j j j 1 f = j = 0 otherwse j k k kj It can be shown that T M = ( I W ) ( I W )
29 Sparse egenvalue problem Optmal embeddng gven by bottom d+1 egenvectors, correspondng to the d+1 smallest egenvalues (Raylegh-Rtz theorem). Soluton Dscard bottom egenvector [1 1 1] (wth egenvalue zero). Other egenvectors satsfy constrants.
30 Surfaces N=1000 nputs k=8 nearest neghbors
31 Lps N=15960 mages K=24 neghbors D=65664 pxels d=2 (shown)
32 Pose and expresson N=1965 mages k=12 nearest neghbors D=560 pxels d=2 (shown)
33 Propertes of LLE Strengths: Fast No local mnma Non-teratve Non-parametrc (only heurstc s neghbourhood sze). Weaknesses: Senstve to shortcuts No estmate of dmensonalty
34 LLE versus Isomap Many smlartes Graph-based, spectral method No local mnma Essental dfferences Does not estmate dmensonalty No theoretcal guarantees Constructs sparse vs. dense matrx Preserves weghts vs. dstances Much faster
35 Laplacan Egenmaps Map nearby nputs to nearby outputs, where nearness s encoded by graph. Summary of the Algorthm 1. Identfy k-nearest neghbours (as n LLE) 2. Assgn weghts to neghbours 3. Sparse egenvalue problem
36 Step 2. Construct the graph Vertces represent nputs. Undrected edges connect neghbours. Assgn weghts to neghbours: Smple: W j = 1 or Heat kernel W j = exp β ( ) 2 x x j
37 Step 3. Graph Laplacan Compute outputs by mnmzng: = W y Ψ( y) y under approprate constrants Ψ = j j ( ) ( 2 2 y = W y + y 2y y ) y 2 j D + j j y 2 j D jj j j 2 2 j y j y j W j W j s symmetrc = 2y t Ly D = W j j Graph Laplacan L = D W
38 Step 3. Generalzed egenvalue problem Mnmze constraned by y t Ly Dy= 1 Optmal embeddng: y t ( Le = λde) gven by bottom d+1 egenvectors (correspondng to the d+1 smallest egenvalues). Soluton: Dscard bottom egenvector [1 1 1] (wth egenvalue zero). Other egenvectors satsfy constrants.
39 Analyss on Manfolds Consder Remannan manfold a real dfferentable manfold n whch tangent space s equpped wth dot product. Laplace Beltram operator Ω Ω R has a natural operator on dfferentable functons. s a second order dfferental operator defned as a dvergence of the gradent D = 2 x 2
40 Spectral desomposton of Assume L 2 (Ω) s space of all square ntegrable functons on Ω s a self-adjont postve sem- defnate operator and ts egenfunctons form the bass. Thus all f n L 2 (Ω) can be wrtten as ( x) = α e ( x) (provded Ω s compact) f
41 Smoothness functonal Defned as value close to zero mples f beng smooth. ( ) Ω Ω = = = 2, ) ( 2 L f f d f f d f f S ω ω value close to zero mples f beng smooth. Snce we have e e e S λ = =, ) ( = = = e e f f f S α λ α α,, ) ( choosng the lowest p egenfunctons provdes a maxmally smooth approxmaton to the manfold.
42 Spectral graph theory Weghted graph s dscretzed representaton of manfold. Laplacan measures smoothness of functons over manfold and graph. Manfold: Graph: Ω j f W 2 dω = f f dω ( ) 2 t f f f Lf j j =
43 Interpretng Laplacan Egenmaps Egenvectors functons from nodes to R n a way that "close by" ponts are assgned "close by" values. Egenvalues measure how close are the values of neghbourng ponts smoothness.
44 Example: S1 (the crcle) Contnuous Egenfunctons of Laplacan are bass for perodc functons on crcle, ordered by smoothness. Egenvalues measure smoothness.
45 Example: S1 (the crcle) Dscrete (n equally spaced ponts) Egenvectors of graph Laplacan are dscrete snes and cosnes. Egenvalues measure smoothness.
46 Laplacan vs LLE More smlar than dfferent Graph-based, spectral method Sparse egenvalue problem Smlar results n practce Essental dfferences Preserves localty vs local lnearty Uses graph Laplacan
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