Patent Application: US-15099408-A

Abstract:
a method of partitioning a weighted combinatorial graph representative of a dataset consists of the steps of generating a generalized laplacian matrix corresponding to the combinatorial graph , computing the eigenstructure of the generalized laplacian matrix , determining if an end criterion is satisfied using the eigenstructure , and if the end criterion is not satisfied , calculating new values for at least some of the plurality of weighting factors using the eigenstructure , updating the combinatorial graph with the new values for at least some of the weighting factors , and returning to the generating step .

Description:
turning to fig2 , a diagram illustrating the disclosed method according to one embodiment is illustrated . in fig2 , the process starts with the receipt of an input dataset at 20 . as mentioned in the background , the input dataset could be representative of audio signals , images , volumetric data , etc . at 22 , a combinatorial weighted graph is constructed from the dataset using known techniques . the particular technique used to construct the combinatorial weighted graph is not important to the practicing of the disclosed spectral rounding method . at 24 , the laplacian matrix corresponding to the combinatorial graph is generated . the particular technique used to generate the laplacian matrix is not important to the practicing of the disclosed spectral rounding method . after the laplacian matrix is generated , the eigenstructure , i . e ., the eigenvalues and eigenvectors , of the laplacian matrix are computed . although the particular method used to calculate the eigenvalues and eigenvectors is not important to the practicing of the disclosed spectral rounding method , we prefer to use the method of computing the generalized eigenstructure disclosed in i . koutis , g . miller , “ a linear work , o ( n ^{ ⅙ }) parallel time algorithm for solving planar laplacians ,” soda : siam symposium on discrete algorithms ( soda 2007 ), which is hereby incorporated by reference for all purposes . after the generalized eigenstructure of the laplacian matrix is computed , the eigenvalues are examined to determine if an end criterion has been satisfied at 28 . various different end criterion may be used . for example , one end criterion is to determine if the eigenvalue has reached zero . another end criterion could include determining if the reweighting and iteration steps show no further improvement . yet another end criterion could involve some subjective user input as to what is considered to be a satisfactory solution . those of ordinary skill in the art will recognize that many different end criterion may be used . if at 28 the end criterion is satisfied , the process terminates ( ends ). if at 28 the end criterion is not satisfied , the process continues at 30 with the calculation of new weighting factors using the eigenvectors as a starting point . the calculation of the new weighting factors is described in detail below . the new weighting factors are used at 32 to update the combinatorial graph . the process then returns via loop 34 to step 24 from which the process repeats until the end criterion is satisfied . the disclosed spectral rounding method may be practiced on a wide variety of hardware configurations . one example of such a hardware configuration is illustrated in fig3 . in fig3 , a computer 40 is shown having various input devices such as keyboard 42 , mouse 44 , microphone 46 , among others . the computer is also shown having various output devices such as headphones 48 and monitor 50 , among others . the disclosed spectral rounding algorithm may be carried on the hard drive of the computer 40 , loaded onto the computer 40 via a computer readable disc inserted into one of the computer &# 39 ; s disc drives 52 , or accessed via a network connection . the reader will understand that the disclosed spectral rounding algorithm may be practiced on computer 40 alone , or as part of a parallel processing network in which computer 40 is one node . the generality of the disclosed spectral rounding algorithm is not intended to be limited by the particular hardware configuration used to practice the disclosed method . the present disclosure relates to weighted graphs of the form g =( v , e , w ). there is a natural isomorphism between graphs and their laplacians . throughout this paper we let g =( v , e , w ) denote an edge weighted undirected graph without multiple edges or self loops , where v is a set of n vertices numbered from 1 to n , e is a set of m edges , and w : e →[ 0 , 1 ] is the edge weighting . we associate four matrices with the graph : first , w the weighted adjacency matrix , w ij = { w ij = w ⁡ ( i , j ) if ⁢ ⁢ ( i , j ) ∈ e 0 otherwise the weighted degree of vertex i is d i = σ j = 1 w ij . . we assume that no vertex has zero degree . d ij = { d i if ⁢ ⁢ i = j 0 otherwise . third , the generalized laplacian or simply the laplacian of g is l = d − w and the fourth is the normalized laplacian { circumflex over ( l )}= d − 1 / 2 ld − 1 / 2 . a valuation function of the vertices of g is a function ƒ : v → , that is , a map that assigns a real number to each of the vertices . a vector in the nullspace of ( l , d ) is a valuation such that lƒ i = 0 · dƒ i . the dimensionality of the nullspace is the number of d - orthogonal vectors ƒ i t dƒ j = 0 satisfying the nullspace equality . note , that the number of connected components in g is equal to the dimensionality of the nullspace of ( l , d ). the generalized eigenvalues and eigenvectors of ( l , d ) are the solutions of lƒ = λdƒ . in this case the normalized rayleigh quotient of the valuation ƒ is ƒ t lƒ / ƒ t dƒ . the generalized eigenvalues of ( l , d ) are all real numbers and bound from below by λ 1 = 0 , by convention we order the generalized eigenvalues of ( l , d ) as 0 = λ 1 ≦ λ 2 ≦ . . . ≦ λ n . we make a simple , but important , observation about the rayleigh quotient ƒ t lƒ / ƒ t dƒ : given a weighted symmetric graph g =( v , e , w ) then the normalized rayleigh quotient can be written a : f t ⁢ lf f t ⁢ df = ∑ ( i , j ) ∈ e , i & lt ; j ⁢ ( f i - f j ) 2 ⁢ w ij ∑ ( i , j ) ∈ e , i & lt ; j ⁢ ( ( f i ) 2 + ( f j ) 2 ) ⁢ w ij ( 3 ) where ƒ 1 = ƒf ( ν i ). the main importance of the above equation is that for each valuation ƒ and each edge e ij in e we get the fraction ( f i - f j ) ⁢ 2 ( f i ) 2 + ( f j ) 2 . we further note that the above rayleigh quotient computes the weighted mediant of this set of fractions . is the weighted mediant . note that the mediant is distinct from the geometric mean of a set of fractions . however , the mediant shares numerous properties with the geometric mean , for example , the value of the mediant lies between the largest a quotient cut of the graph g =( v , e , w ) attempts to minimize the ratio of cost of the cut ( i . e ., the sum of the weights associated with the edges whose removal disconnects the graph ) and the size of the resulting partitions . in this way it seeks to balance the size of each sub - graph ( proportional to the total weight in the sub - graph ) with the cost of the cut . a graph is taken to be an expander graph if its quotient cut function is bound from below by a constant . the partition indicator matrix ( pim ) associated with a cut is an ( n × k ) matrix with a single one in each row and column . note , for a pim p that the quotient cut cost is proportional to trace (( p t dp ) − 1 ( p t lp )). the relaxation of the search over matrices p minimizing this function to continuous vectors is one of the early derivations of eigenstructure - based approaches [ 1 ]. as described in the summary section , the second step of the method is the computation of k ′ generalized eigenpairs with small value . the generalized eigenvectors ƒ i and eigenvalues λ i of the matrix pencil ( l , d ) are the solutions to the equation by convention , the generalized eigenvalues λ i are ordered such that 0 = λ 1 ≦ . . . ≦ λ n n . the extremal values of ( a , d ) are computed using existing software from the public domain for sparse numerical linear algebra packages [ 6 ], ( i . e ., lapack ). the manner in which the generalized eigenvector associated with the smallest k ′ generalized eigenvalues are employed by the disclosed spectral rounding method to effect a partitioned graph is described in the next . as described in the summary section , the third step of the method determines a suitable reweighting of the graph g such that repeated application of the reweighting disconnects the graph . this begins with the necessary condition for decreasing linear combinations of generalized eigenvalues of the matrix pencil ( l , d ). the condition depends on a set of novel observations in relation to rounding the generalized eigenvectors of ( l , d ) to obtain a partitioning of the graph . in particular , we note that the dimensionality of the nullspace of l ′ is equal to the number of connected components in g under the weighting w ′. in particular , for a fixed valuation function ƒ : v → , there exists a reweighting w ′ of the graph g =( v , e , w ) such that where ( l ′, d ′) is the matrix pair on g =( v , e , w ′) corresponding to the reweighting w ′. the above inequality will be referred to as the rayleigh quotient constraint in subsequent sections . next it is shown that , if the vector ƒ is a generalized eigenvector of the matrix pair ( l , d ), the eigenvalue is decreased as well . given a weighted graph g =( v , e , w ), matrices l and d , the simple eigenpair ( ƒ , λ )| lƒ = λdƒ , and a new weighting w ′ such that f t ⁢ l ′ ⁢ f f t ⁢ d ′ ⁢ f & lt ; f t ⁢ lf fdf = λ then the derivative of the generalized eigenvalue function λ ( t ) of the reweighting matrix curve w ( t )= w + tw ′ is well defined for small t and as w ( 0 )= w and thus l ( 0 )= l , d ( 0 )= d by definition . the bound on on is deduced from the observation that this derivative may be written as thus , if the rayleigh quotient constraint holds for the reweighting , the eigenvalue derivative is negative , and the associated eigenvalue must decrease along the matrix curve over the reweighting . these observations have been generalized to multiple eigenvalues by the inventors in reference [ 18 , 17 ]. two example reweighting schemes , that provide reweightings satisfying the rayleigh quotient constraint , are given below . those skilled in the art can development many more valid reweighting schemes that satisfy the rayleigh quotient constraint . the first concerns reweighting a single valuation and the second concerns reweighting multiple valuations simultaneously . the first example , proposed in [ 17 ], is proportionality to the reciprocal of the edge fractions r ij ⁢ ( f i - f j ) 2 ( f i 2 + f j 2 . that is , the new weighting w ′ ij = w ij · g ( r ij − 1 ). where g is any monotone function mapping the interval [ 0 , ∞] to [ 0 , 1 ] ( for example , the stereographic projection or 1 − e − c · r ij − 1 ). recall that a function is monotone over a set , if it is order preserving . intuitively , the reciprocal reweighting applies large weight to the smallest edge fractions and small weight to the edge fractions with large value . this simple reweighting can be applied to an arbitrary eigenpair ( ƒ k , λ k ) satisfying lƒ k = λ k dƒ k . through a basic property of linear algebra driving λ k to zero ensures that λ 1 = . . . = λ k −− 1 = λ k = 0 . the reweighting where λ k = 0 , ensures that the dimensionality of the nullspace of ( l ′, d ′) is k , thus the graph is partitioned into k pieces , and the termination condition ( i . e ., end criterion ) for the method is achieved . the second concerns reweighting using multiple eigenvectors . for simplicity , the example only concerns two but trivially generalizes to more eigenvectors . as an intuition , using multiple eigenvectors result in a mediant of mediants calculation , and therefore , the rayleigh quotient constraint carries through to multiple valuations . for a weighted graph g =( v , e , w ) with matrices l and d and simple eigenpairs ( ƒ , λ ƒ )| lƒ = λd ƒ and ( g , λ g )| lg = λ g d g , given a reweighting w ′ such that 1 λ f ⁢ f t ⁢ l ′ ⁢ f + 1 λ g ⁢ g ′ ⁢ l ′ ⁢ g f t ⁢ d ′ ⁢ f + g t ⁢ d ′ ⁢ g & lt ; 1 λ f ⁢ f t ⁢ l ⁢ ⁢ f + 1 λ g ⁢ g t ⁢ l ⁢ ⁢ g f t ⁢ df + g t ⁢ d g = 1 then ⅆ λ f ⁡ ( t ) ⅆ t + ⅆ λ g ⁡ ( t ) ⅆ t & lt ; 0 at t = 0 . we begin by stating a related quantity of interest , the derivative of the fractional average of rayleigh quotients on ƒ and g for the matrix curve w = w + t · w ′ as : ⅆ ⅆ t [ 1 λ ⁢ f t ⁡ ( t ) ⁢ l ⁡ ( t ) ⁢ f ⁡ ( t ) + 1 λ ⁢ g t ⁡ ( t ) ⁢ l ⁡ ( t ) ⁢ g ⁡ ( t ) f t ⁡ ( t ) ⁢ d ⁡ ( t ) ⁢ f ⁡ ( t ) + g t ⁡ ( t ) ⁢ d ⁡ ( t ) ⁢ g ⁡ ( t ) ] and examine its derivative centered at t ,= 0 . first , we must fix the scale of the eigenvectors ƒ ( t ) and g ( t ), we choose ƒ ( t ) t d ( t ) ƒ ( t )= g ( t ) t d ( t ) g ( t )= 1 w . l . o . g . thus , equation 5 simplifies to ⅆ ⅆ t [ 1 λ ⁢ f t ⁡ ( t ) ⁢ l ⁡ ( t ) ⁢ f ⁡ ( t ) + 1 λ g ⁢ g t ⁡ ( t ) ⁢ l ⁡ ( t ) ⁢ g ⁡ ( t ) 1 + 1 ] = 1 2 ⁢ ( 1 λ f ⁢ ⅆ ⅆ t ⁢ λ f ⁡ ( t ) + 1 λ g ⁢ ⅆ ⅆ t ⁢ λ g ⁡ ( t ) ) by the linearity of the derivative . we may now substitute the functional form of 1 2 ⁢ ( 1 λ f ⁢ f t ⁡ ( l ′ - λ f ⁢ d ′ ) ⁢ f + 1 λ g ⁢ g t ⁡ ( l ′ - λ g ⁢ d ′ ) ⁢ g ) 1 2 ⁢ ( 1 λ f ⁢ f t ⁢ l ′ ⁢ f - λ f λ f ⁢ f t ⁢ d ′ ⁢ f + 1 λ g ⁢ g t ⁢ l ′ ⁢ g - λ g λ g ⁢ g t ⁢ d ′ ⁢ g ) & gt ; 0 ( 1 λ f ⁢ f t ⁢ l ′ ⁢ f - f t ⁢ d ′ ⁢ f + 1 λ g ⁢ g t ⁢ l ′ ⁢ g - g t ⁢ d ′ ⁢ g ) & lt ; 0 1 λ f ⁢ f t ⁢ l ′ ⁢ f + 1 λ ⁢ ⁢ g ⁢ g t ⁢ l ′ ⁢ g & lt ; f t ⁢ d ′ ⁢ f + g t ⁢ d ′ ⁢ g the previous section detailed the constraints on valid reweighting and exhibited two valid procedures . this section concerns the fourth and final step of the invention , the application of and iteration over reweightings of g . after fixing a reweighting w ′, using a valid procedure , a line search is performed along the matrix curve w ( t )= w + tw ′ to determine the setting of t , t *, at which the minimum value of σ i = 1 n α i λ i ( t ), where α i ≧ and σ i = 1 n α i = 1 for a fixed α , is achieved . the combined weighting of the graph , w ( t *)= w + t * w ′, is taken as the new weighting of g and the process is repeated until termination . this is termed application of a reweighting . the successful termination of iteration is guaranteed by the following two observations , given here in a simplified form . first , a reweighting procedure , consistent with the rayleigh quotient constraint , can always decrease the eigenvalues ( thus has not reached a fixed point ). this is formally characterized by noting that w ′= w · w r , where w r a valid reweighting of the edges . the fixed point condition of reweighting thus requires that w r ( e ) ε 0 , 1 for all edges eε e ( g ). under fixed - point conditions , the repeated application of the new weighting does not change the generalized eigenvalues of g . this only occurs when the graph is disconnected or when all the edge fractions are bound from below by 1 . by the convexity property of mediants , the smallest eigenvalue λ of ( l , d ) is bound from below by 1 , thus the quotient cut is bound by a constant due to the cheeger inequality and the graph is an expander . therefore , the disclosed method has only fixed points at the proper termination conditions for all graphs excluding a set of expanders . the second observation , a standard result in real analysis ( heine - borrel theorem ), dictates that a sub - sequence of the sequence generated by the iteration must converge to a fixed point , as the sets in question : weights , eigenvalues , and eigenvectors , are all compact and closed . therefore , the method converges and terminates , ensuring that a partitioning of the graph is achieved by applying the method above . the application of the disclosed method to the digital signal segmentation problem requires the construction of a combinatorial graph from a digital signal . these signals could come from audio streams , images , video voxels or arbitrarily high dimensional medical or scientific scans . the goal of digital signal segmentation is to segregate a signal into collections of highly coherent components . for example , in audio processing the components might correspond to samples composing the words or syllables in the sequence . in image processing the components correspond to compact areas of the image that share a measurable quality such as color , texture , or a common boundary . the construction of a weighted graph g =( v , e , w ), where v is the vertex set , e the edge set and w a weighting of e from a digital spatial signal , is a three - stage process . the resolution , isotropic or otherwise , must be established . this stage assigns samples to vertices in the graph and ultimately determines the size of the graph . the second stage establishes the topology of the graph g . this step determines the members of the edge set e of g , two vertices u and v in v are said to be adjacent if the edge ( uv ) is in e . the third stage performs a sequence of comparisons between adjacent vertices based on the signal samples associated with the vertices comprising the edge . this framework procedure produces a weighted combinatorial graph g from a digital signal — the segmentation of which can then be accomplished by partitioning g using the computation method described above . the construction is as follows : a vertex in g for each element in the digital signal — for example if the digital signal is derived from a standard digital photograph , each pixel in the image will correspond to a signal vertex in v . the vertices are then connected to neighboring samples by edges in e which are weighted in proportion to the similarity between elements at the observed sites . for example , in the case of the standard 4 - connected mesh of a 2d image , this comprises approximately 4 comparisons per pixel , as each pixel is compared to its neighbors to the north , south , east , and west . the measured level of similarity is then used to weight the corresponding edge in g . there are numerous other applications that require the construction of a graph from supporting data . these include social network analysis ( finding communities in a social network ), in which the graph edges and weights might correspond to the frequency of emails between parties . clustering of amino acids in proteins in which proximity and the chemical properties of amino acid pairs determine the connectivity and weighting of the graph . these as well as numerous other applications of the reduction to graph partitioning all benefit from the partitioning method disclosed above . the parameters used in constructing a weighted graph from an image were fixed for all the results presented in this section . the graph g =( v , e , w ) represents an image as follows . for each pixel in the image , a vertex in v is assigned . if two pixels are connected in e a weight in w is determined based on the image data . the graph connectivity , e , was generated by connecting pixels to 15 % of their neighboring pixels in a 10 pixel radius . the initial weighting w of the graph g =( v , e , w ) was determined using the intervening contour cue described in [ 15 ]. this cue assigns small weights to pixels that lie on opposite sides of a strong image boundary , and large weights otherwise . we compiled a set of 100 images from google images using the keywords farm , sports , flowers , mountains , and pets . examples from this data set , and segmentations , can be found in fig4 . we note that changes in the cut value often correlate with large changes in the co - membership relationships on the image pixels . to quantitatively compare the methods on natural images , we report the divergence distance and ncut improvement factor c . table 1 below is a comparison between the disclosed spectral rounding sr method and the multiway cut algorithm of yu and shi [ 20 ] eig on segmentations of natural images . the average cluster entropy over sr - segmentations of the image collection is 1 . 62 ± 0 . 4 . additional examples are found in tolliver , et al . “ graph partitioning by spectral rounding : applications in image segmentation and clustering ” appearing in computer vision and pattern recognition 2006 : pp 1053 - 1060 , which is hereby incorporated by reference for all purposes . k . anstreicher and h . wolkowicz . on lagrangian relaxation of quadratic matrix constraints . siam journal matrix analysis applications , 22 ( 1 ): 44 - 55 , 2000 . 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[ 16 ] d . spielman and s . h . teng . nearly linear time algorithms for graph partitioning , graph sparsification , and solving linear systems . stoc , pages 81 - 90 , 2004 . [ 17 ] d . a . tolliver and g . l . miller . graph partitioning by spectral rounding : applications in image segmentation and clustering . ieee computer vision and pattern recognition , pages 1053 - 1060 , june 2006 . [ 18 ] david tolliver . spectral rounding & amp ; image segmentation . phd thesis , robotics institute , carnegie mellon university , pittsburgh , pa ., august 2006 . [ 19 ] e . p . xing and m . i . jordan . on semidefinte relaxtion for normalized k - cut and connections to spectral clustering . tr - csd - 03 - 1265 , university of california berkeley , 2003 . [ 20 ] s . yu and j . shi . multiclass spectral clustering . in iccv , october 2003 . while the present invention has been disclosed in conjunction with preferred embodiments thereof , those of ordinary skill in the art will recognize that many modifications and variations are possible . the present document is intended to be limited only by the scope of the following claims .