Evaluation of Space Partitioning Data Structures for Nonlinear Mapping

Transcription

1 WSCG 2015 Conference on Computer Graphcs, Vsualzaton and Computer Vson Evaluaton of Space Parttonng Data Structures for Nonlnear Mappng Myasnov E.V. Samara State Aerospace Unversty, Image Processng Systems Insttute of the Russan Academy of Scences Samara, Russa ABSTRACT Nonlnear mappng (Sammon mappng s a nonlnear dmensonalty reducton technque operatng on the data structure preservng prncple. Several possble space parttonng data structures (vp-trees, d-trees and cluster trees are appled n the paper to mprove the effcency of the nonlnear mappng algorthm. At the frst step specfed structures partton the nput multdmensonal space, at the second step space parttonng structure s used to buld up the lst of reference nodes used to approxmate calculatons. The further steps perform ntalzaton and teratve refnement of the low-dmensonal coordnates of obects n the output space usng created lsts of reference nodes. Analyzed space parttonng data structures are evaluated n terms of the data mappng error and runtme. The experments are carred out on the well-nown datasets. Keywords Dmensonalty reducton, nonlnear mappng, Sammon mappng, multdmensonal scalng, MDS, space parttonng, space decomposton, vp-tree, d-tree, cluster tree INTRODUCTION Nonlnear mappng algorthm [Sam69] also nown as a Sammon mappng s one of the most well-nown exploratve data analyss technques. It belongs to a more broad class of nonlnear dmensonalty reducton technques operatng on the data structure preservng prncple. Nonlnear mappng s wdely used n scentfc research, and n many areas of producton actvtes. In sgnal and mage analyss nonlnear mappng has been appled n creaton of navgaton systems for mage and multmeda collectons. For example, for dgtal mage collectons feature nformaton based on vsual characterstcs of mages s extracted at frst. Then the nonlnear mappng algorthm s used to map the mages from multdmensonal feature space to the navgaton space (2 or 3 dmensonal. Smlar mages are placed close to each other n ths Permsson to mae dgtal or hard copes of all or part of ths wor for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. navgaton space and a user can browse the collecton due to vsual characterstcs of mages. Another applcaton of nonlnear mappng s automated segmentaton and thematc classfcaton of multspectral satellte mage. In ths case separate pxels are frst clusterzed due to ts feature nformaton and then the clusters are mapped nto two-dmensonal space usng nonlnear mappng algorthm. In ths space smlar clusters are placed close to each other that allows user to merge clusters belongng to same segments. Such merged clusters can be later semantcally labeled to perform thematc classfcaton of an mage. Nonlnear mappng performs proecton from some nput multdmensonal space to output lowdmensonal space for a gven set of obects (are often referred to as data ponts O { o1, o2,..., on}. We consder the preservaton of the data structure as the preservaton of the parwse dstances between the obects. That s the dstances d ( o, o between pars of obects o and o n the output low-dmensonal space should approxmate correspondng dstances d o, o n the nput multdmensonal space. It s ( obvous that such a proecton cannot preserve dstances exactly and the followng data Full Papers Proceedngs 109 ISBN

2 WSCG 2015 Conference on Computer Graphcs, Vsualzaton and Computer Vson representaton error allows to estmate the qualty of the mappng: o, o O o, o O 1 o, o d ( o, o (1 o, o o, o Applyng gradent descent approach to mnmze the data representaton error we obtan the followng teratve equaton for coordnates y of obect o n the output low-dmensonal space: where y ( t 1 y ( t m ς( o, o (2 m o, o O 2 o O o, o o, o d ( o, o ς( o, o y o, o d ( o, o, y 2 (3 Here t s the number of teraton, s some coeffcent that affects the convergence of the algorthm. The notatons n the above equatons are dfferent from the ones usually used n lterature to smplfy the further descrpton. An example of a nonlnear mappng for a synthetc dataset s shown on fg. 1. The dataset conssts of a set of ponts obtaned from the text depcted on a cylnder n 3D space (fg. 1.a. Fg 1.b shows the results obtaned usng the prncpal component analyss (PCA. Dataset s mapped on the frst two prncpal components. The results of the descrbed above nonlnear mappng algorthm s shown on the fg. 1.c. As t can be seen the local data structure appears. Fg. 2 shows a mappng of a set of mult-spectral values (4 spectral bands of pxels n 3x3 neghborhoods (total 36 attrbutes n a LANDSAT satellte mage [LAN]. Dfferent classes are shown n dfferent color. To obtan a soluton one should ntalze the coordnates y (0 and teratvely refne coordnates of all the obects n accordance to (2 untl the coordnates become stable. Unfortunately, ths smple teratve procedure gven above has a sgnfcant drawbac. If the set O contans N obects then the computatonal complexty s O(N 2 per teraton (the computatonal complexty for the data representaton error s roughly the same. (a (b (c Fgure 1. Mappng for synthetc dataset: (a synthetc 3D data; (b mappng on the frst two prncpal components; (c nonlnear mappng (data representaton error s = The algorthm becomes tme consumng for relatvely small sets of obects contanng hundreds and thousands of obects. The problem becomes more sgnfcant for sets contanng some tens thousands of obects as the memory needed to store precomputed dstances between the obects n the nput multdmensonal space s N(N-1/2 (havng account that the dstance matrx s symmetrcal. For example, for a set of obects we need about 1,5 Gb of operatonal memory to store the dstances as a 8 byte floatng pont values. Recomputng the dstances maes the optmzaton process even more tme consumng and dependent on the dmensonalty of the nput space. As an example, the tme needed to compute the error n accordance to (1 for a MNIST dataset contanng obects used n ths paper for an expermental study taes more than one hour. Full Papers Proceedngs 110 ISBN

3 WSCG 2015 Conference on Computer Graphcs, Vsualzaton and Computer Vson Fgure 2. Nonlnear mappng for the Landsat Satellte dataset (data representaton error s = One of the most effectve and promsng approaches to reduce the computatonal complexty s to perform herarchcal parttonng of space. In the case of the consdered problem such parttonng can be performed usng dfferent space parttonng structures. Ths study s devoted to the comparson of several space parttonng structures n terms of the qualty of mappng, as well as n terms of operatng tme. Related wors To address the problem of a computatonal complexty of the nonlnear mappng the number of technques has been proposed. For ths purpose the trangulaton [Lee77], and lnear transformaton [Pe99] are used. Later the performance of the nonlnear mappng has been mproved by extendng data representaton error usng left [Sun11a] and rght [Sun11b] Bregman dvergence. Tang nto account that nonlnear mappng belongs to the multdmensonal scalng technques t s worth mentonng the methods based on the stochastc optmzaton [Cha96]. The dea of a herarchcal space parttonng to reduce the computatonal complexty has orgnated from physcs where t s wdely used n modellng systems consstng of a huge number of obects (n-body smulatons. The parttonng of the space has been mplemented n graph drawng, for example, n [Fru91] (regular decomposton and [Qu01] (herarchcal decomposton to approxmate the forces actng on the vertces of the graph. In dmensonalty reducton the herarchcal parttonng of the nput multdmensonal space smlar to Barnes-Hut [Bar86] algorthm has been mplemented n [Mya12] to mprove nonlnear mappng. The herarchcal parttonng of the output low-dmensonal space usng Barnes-Hut algorthm has been mplemented n [Van13, Yan13] to accelerate stochastc neghbor embeddng algorthm (t-sne. In [Vla14] another wdely used fast multpole method [Gre87] has been appled to speed up elastc embeddng algorthm. Approxmate computatons The man dea of speedng up computatons usng the space parttonng technques s to dvde the whole set of obects O to some subsets s,... so that all 1, s 2 obects o s from some subset s should possess smlar characterstcs. That s all obects from some subset s are stuated close to each other n the gven space. In ths case obects n a subset s can be analyzed not ndvdually, but as a sngle obect under some crcumstances. Assume that the subset s s stuated far from the obect summaton o s o, o d ( o, o y o, o d ( o, o can be approxmated as follows s o, s d ( o, s y y o, s d ( o, s o. Then the exact y where s s the number of obects n the consdered subset, d o, s s the dstance from the obect o to ( the center of the subset s s n the nput space, d ( o, s s the dstance from the obect o to the center of the subset n the output space, y s s the coordnates of the center of the group n the output space. Now f all the obects of the orgnal set O are dvded nto subsets s S, then (2 taes the form y ( t 1 y ( t m ~ ς( o, s, (4 ~ ς( o, s s s S o, s d ( o, s y o, s d ( o, s y s, (5 It s obvous that the equaton (4 allows to approxmate (2 wth a certan accuracy that depends on how well obects are dvded nto subsets. For ths reason, there s a queston about how to perform such a decomposton. Full Papers Proceedngs 111 ISBN

4 WSCG 2015 Conference on Computer Graphcs, Vsualzaton and Computer Vson Descrpton of the Methods The base scheme for the method descrbed below s taen from [Mya12] wth some modfcatons. In general the consdered method conssts of the four followng steps: 1. Parttonng of the nput space 2. Constructon of reference nodes lsts 3. Intalzaton n the output space 4. Iteratve optmzaton The above stages are dscussed below n more detal. Parttonng of the nput space The frst step of the orgnal method assumed the herarchcal clusterng to partton the nput space. Ths lead to the O(N 2 complexty n the case of effectve agglomeratve clusterng. In the case of dvsve method based on the neural networ (WTA that was mplemented n [Mya12] the runtme of herarchcal clusterng s hghly dependent on the parameters of the algorthm. At the same tme there are a number of space parttonng trees that can be bult n less tme and that used wdely n multmeda databases, geographc nformaton systems, nformaton retreval, computer graphcs and so on. The revew of such data structures s beyond the scope of ths paper. A comprehensve wor on space parttonng trees can be found, for example, n [Sam06]. In ths wor several bnary space parttonng trees were used and compared for nput space parttonng: d-tree [Ben75], metrc tree (vp-tree [Uhl91, Ya93], and bnary tree based on the smplfcaton of mnmax dstance clusterng approach [Tou74] (further cluster tree. Such choce s motvated to study structures that are dfferent n ther propertes. Kd-tree s one of the old and well-nown space parttonng structures based on the recursve splttng a space wth hyperplanes orthogonal to the coordnate axes. Vp-tree s another well-nown space parttonng structure based on the hypershercal parttonng of a space. Cluster tree s the dstance based structure that parttons a space wth hyperplanes orthogonal to a par of chosen dstant ponts. All the mentoned structures can be bult n O(N log N operatons (f the tree s balanced that s more sutable for the consdered tas. As the constructon algorthms for the frst two structures are well nown only the algorthm for latter structure s gven below by a pseudo code. Functon CreateTree( SetOfObects O returns Node begn Create new node S f Number of obects n O s less than threshold then begn S.Chldren = O return S end Fnd mean vector value n O Fnd o1 obect farthest from mean Fnd o2 obect farthest from o1 Create new SetOfObects s1 Create new SetOfObects s2 for each obect o n O begn f ( o1, o < o2, o add o to s1 else add o to s2 end Node n1 = CreateTree( s1 Node n2 = CreateTree( s2 add n1 to S.Chldren add n2 to S.Chldren return S end Let us assume that usng some parttonng method (e.g. usng functon gven above we get the tree-le data structure (fg. 3, contanng the obects of the ntal set O as leaves (termnal nodes. Fgure 3. The structure of a bnary space parttonng tree The tree created at the frst step of an algorthm can be used n optmzaton process mmedately n the followng way. We teratvely update lowdmensonal coordnates for each obect o O n accordance to (4, usng top-level nodes of the tree f such nodes are far enough from the obect o, and low-level nodes of the tree or the obects of the ntal set O n the other case. However, the constructed tree s not used drectly n the optmzaton process n ths paper. Instead of t the specal data structure s used n optmzaton, whch s descrbed n the next subsecton. The detals of the optmzaton process are gven n Iteratve optmzaton subsectons. Constructon of reference nodes lsts It s worth notng that as the parttonng s performed n the nput multdmensonal space then the Full Papers Proceedngs 112 ISBN

5 WSCG 2015 Conference on Computer Graphcs, Vsualzaton and Computer Vson parttonng tree s constant n the optmzaton process. That s why n [Mya12] t was proposed to calculate and to store the set of nodes and the obects whch are nvolved n the optmzaton process (called the lst of reference nodes for each obect o O of the ntal set. The use of the lsts of reference nodes requres addtonal memory but allows to avod recalculatng the parttonng crteron durng the optmzaton process and allows to store precomputed dstances to reference nodes that maes the optmzaton process ndependent on the nput dmensonalty. The straghtforward algorthm for creaton of reference nodes s recursve and descrbed below by pseudo code. Procedure CreateRefLst( Obect o, Node s, RefNodeLst rl begn f o far from s then add s to rl else f s contans obects then for each obect a n s add a to rl else / s contans subtrees / for each node d contaned n s CreateRefLst( o, d, rl end To avod recursve calls the teratve mplementaton usng specal ponters n tree nodes was used n the experments. The descrbed above algorthm s slghtly dfferent compared to [Mya12] by removng the noton of ncomplete node. The use of ncomplete nodes allow to slghtly accelerate calculatons by aggregatng some porton of obects n decomposed nodes of the tree but t requres extra memory to store nformaton about aggregated obects. The frst condton o s far from s defnes the decomposton crteron of the gven node s wth respect to the obect o. Dfferent decomposton crterons were descrbed n physcs lterature (e.g. [Sal94]. And here we adapt (as n [Mya12] smple decomposton crteron based on the radus R of the gven node s. That s we decompose the node of the space parttonng tree under the followng condton: s, o / R( s T where T s a predefned threshold parameter. Otherwse we add the node s of the tree to the lst of the reference nodes. Fg. 4 llustrates ths process. The node s 1 wll be dvded nto two nodes f s1, o / R( s1 T. The node s 2 wll be consdered as a sngle node f d s, o / R( s T ( 2 2. Fgure 4. Illustraton to the constructon of reference nodes lst It s worth notng that the number of the nodes n the lst of the reference nodes s dependent on the dstrbuton of the data. Assumng that the lst of reference nodes contans on average L nodes the memory needed to store the lsts s O[LN]. Intalzaton n the output space The thrd step of the orgnal method performs an ntalzaton of the low-dmensonal coordnates of the obects. Only two frst prncpal components are computed to ntalze obects n the output space and then nput multdmensonal coordnates are proected to the plane formed by these components. Ths approach allows to reduce the computaton tme compared to random ntalzaton as the next step of teratve optmzaton process starts wth the better ntal condtons. Although prncpal components fndng by the covarance matrx s dependent on the dmensonalty of the nput space D and can be estmated as O(D 2 per teraton the overall process can be tme consumng as covarance matrx fndng s O(ND 2 and depends not only on the dmensonalty but on the number N of obects also. To reduce the computatonal tme we use only a small subset of randomly selected obects of the ntal set O mang the overall complexty ndependent on the number of obects. There are other solutons, e.g. use of neural networ approach based on generalzed Hebban (Sanger learnng rule [San89] used n [Mya12]. Iteratve optmzaton After ntalzng low-dmensonal coordnates of obects the optmzaton procedure s performed to fnd sub optmal coordnates of obects n output lowdmensonal space. It conssts of teratve refnement of output coordnates of all obects n accordance wth y ( t 1 y ( t m ~ ς( o, s s S Full Papers Proceedngs 113 ISBN

6 WSCG 2015 Conference on Computer Graphcs, Vsualzaton and Computer Vson where set S s the lst of reference nodes for obect o, and subsets s are reference nodes for obect o. The computatonal complexty of the optmzaton stage of the method can be estmated as O(LN on average where L s the average length of lsts of reference nodes. In practce t may be reasonable to control the data mappng error along the process of optmzaton as t can ndcate the dvergence of the process or t may be used n a stop crteron. The computaton complexty of the data mappng error (1 s roughly the same as the complexty of one step of the base verson of the optmzaton process (O[N 2 ]. Usng the lsts of reference nodes data mappng error can be estmated n O(LN on the average: ~ oo s S 1 o, s oo s S o, s d o, s ( o, s It s worth notng that the above equaton may gve greatly underestmated values of the data mappng error especally n case when approxmaton based on the nput multdmensonal nformaton becomes too coarse. For example, ths can tae place due to poor confguraton of obects n the low-dmensonal space. In the present wor ths equaton was used to control the optmzaton process. When the estmated error value was ncreasng then the coeffcent was decreasng untl the process became convergent. Also ths equaton was used to stop the optmzaton process when the relatve decrease n estmated error for a gven number of teratons dd not exceed the predefned value. Expermental study Two well-nown datasets were used n the presented study. The frst one s the MNIST database of handwrtten dgts [MNI]. The second one s the Corel Image Features Data Set [COR]. The frst database contans dgtal grayscale mages of handwrtten dgts. The database s dvded n two sets: a tranng set contanng nstances, and test set contanng nstances. Images of the tranng set wth sze 28x28 pxels are treated as vectors n 784-dmensonal space n the experments. The second dataset contans features, calculated from the dgtal mages of the Corel mage collecton ( The Corel Image Features Data Set contans nstances. The followng features have been used n the experments: 2 - color hstograms [Swa91] constructed n the HSV color space. Color space was dvded nto 8 ranges of H and 4 ranges of S. The dmensonalty of the feature space s color moments [St95]. Three features were calculated for each color component: mean, standard devaton, and sewness. The dmensonalty of the feature space s 9. - texture features based on co-occurrence matrces [Har73]. Four co-occurrence features (second angular moment, contrast, nverse dfference moment, and entropy were computed n four drectons (horzontal, vertcal, and two dagonal. The dmensonalty of the feature space s 16. To evaluate the effectveness of the methods a number of characterstcs has been measured and calculated: - buldng tme of the bnary space parttonng tree, - buldng tme of the lst of reference nodes, - length of the lst of reference nodes, - ntalzaton tme of the low-dmensonal coordnates, - per teraton executon tme of the optmzaton procedure, - multdmensonal data representaton error. All the descrbed methods were mplemented n C++. The studes wth the MNIST dataset have been carred out on PC based on Intel Core CPU 3.2 GHz. The studes wth the COREL dataset have been carred out on laptop based on Intel Core 3 M370 CPU 2.4 GHz. The wor of the methods stopped when the relatve decrease n estmated error for ten teratons dd not exceed In all cases, the dmenson of the target space has been set equal to two (two-dmensonal data mappng. Some results are shown n fg Fg. 5 shows the dependence of the qualtatve and temporal characterstcs on the threshold parameter T at whch the algorthm moves to the chld nodes of the correspondng parttonng structure (metrc (vp- tree, d-tree or cluster tree. As t can be seen from these results, the small values of T (especally T<1 leads to the expected deteroraton of the mappng qualty due to a coarser approxmaton, whch s reflected n the hgher values of the multdmensonal data representaton error (see fg. 5d. The tme t taes to perform a sngle teraton, ncreases wth ncreasng T (see fg. 5c, due to the large number of the processed reference nodes of the correspondng herarchcal structure (see fg. 5b. Full Papers Proceedngs 114 ISBN

7 WSCG 2015 Conference on Computer Graphcs, Vsualzaton and Computer Vson (a (a (b (b (c (c (d Fgure 5. Dependences on the threshold parameter T (MNIST dataset: (a average buldng tme of the lst of reference nodes (LRN; (b average length of the lst of reference nodes; (c average tme per one teraton of the optmzaton process; (d multdmensonal data representaton error Dependences of qualty and temporal characterstcs on the number of obects s shown n the fg. 6 and 7. The qualty of mappng, measured by the multdmensonal data representaton error (see fg. 6d, 7d s wealy dependent on the type of space parttonng structure. At the same tme the average length of lsts of reference nodes s sgnfcantly larger when we use d-tree (see fg. 6b, 7b. Ths confrms that d-trees are poorly suted for multdmensonal data processng. (d Fgure 6. Dependences on the sample sze (MNIST dataset: (a average buldng tme of the lst of reference nodes (LRN; (b average length of the lst of reference nodes; (c average tme per one teraton of the optmzaton process; (d multdmensonal data representaton error Some results obtaned for the base nonlnear mappng algorthm s shown on fg. 8 (logarthmc scale. Tmngs for the case of precomputed dstances are not shown for large sample szes due to memory lmtatons. The multdmensonal data representaton error s shown on fg. 7 for comparson. Full Papers Proceedngs 115 ISBN

8 WSCG 2015 Conference on Computer Graphcs, Vsualzaton and Computer Vson (a (b (c (d Fgure 7. Dependences on the sample sze (COREL dataset: (a average buldng tme of the lst of reference nodes (LRN; (b average length of the lst of reference nodes; (c average tme per one teraton of the optmzaton process; (d multdmensonal data representaton error As we can see the error values for the base method s only slghtly better than error values obtaned usng the studed methods. Note that the experments performed on other datasets descrbed above, show smlar results. Fgure 8. Dependency of the average tme per one teraton on the sample sze for the base algorthm (COREL dataset Some examples of the nonlnear mappng obtaned usng space parttonng structure and the base algorthm for a subset contanng 5000 nstances of the MNIST dataset s shown n fg. 9. An example of the nonlnear mappng usng the descrbed approach for obects of the MNIST database s shown n fg. 10. The database was processed n less than 30 mnutes ncludng error estmaton at each teraton (19 mnutes wthout error estmaton. The multdmensonal data representaton error was equal to Concluson In ths paper, we conducted a study of several space parttonng structures namely metrc trees (vp-trees, d-trees, and cluster trees to speed up the nonlnear mappng. The study showed that the qualty of mappng s wealy dependent on the type of the structure but the average teraton tme was dfferent for the consdered structures. For d-tree the number of reference nodes was sgnfcantly larger than for the other structures, hence the average teraton tme was larger. Thus for the presented data sets metrc trees (vptrees and cluster trees can be effcently appled to partton the nput space for the consdered problem. Usng of the consdered data structures made t possble to generate low-dmensonal embeddngs for relatvely large datasets n a comfortable tme. At the same tme the data representaton error for the base algorthm had slghtly lower values. To mprove the qualty of the mappng one can use the consdered structures ncreasng the threshold parameter T or use the obtaned low-dmensonal confguraton as an ntal confguraton for the base method. ACKNOWLEDGMENTS Ths wor was fnancally supported by the Russan Foundaton for Basc Research, proect a. Full Papers Proceedngs 116 ISBN

9 WSCG 2015 Conference on Computer Graphcs, Vsualzaton and Computer Vson (a (b Fgure 9. Nonlnear mappng for 5000 nstances of MNIST database: (a cluster tree, 207 teratons, threshold T = 1, multdmensonal data representaton error = ; (b base method, 194 teratons, multdmensonal data representaton error = Fgure 10. Nonlnear mappng for tranng set of MNIST database (60,000 examples Full Papers Proceedngs 117 ISBN

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