Sparse-TDA: Sparse Realization of Topological Data Analysis for Multi-Way Classification

3.4 MB·

by  Wei Guo

IEEETRANSACTIONSONKNOWLEDGEANDDATAENGINEERING 1
Sparse-TDA: Sparse Realization of Topological
Data Analysis for Multi-Way Classification
Wei Guo, Student Member, IEEE, Krithika Manohar, Steven L. Brunton and Ashis G. Banerjee, Member, IEEE
Abstract—Topologicaldataanalysis(TDA)hasemergedasone to binary classification have not been particularly successful
of the most promising techniques to reconstruct the unknown either. The success rates diminish further in the absence of a
shapes of high-dimensional spaces from observed data samples.
large number of data samples for each of the labeled classes.
TDA,thus,yieldskeyshapedescriptorsintheformofpersistent
7 The primary reason is that all of these methods encounter
topological features that can be used for any supervised or
1 unsupervised learning task, including multi-way classification. difficulties in selecting the right set of distinguishing features
0
Sparse sampling, on the other hand, provides a highly efficient among the different classes.
2
technique to reconstruct signals in the spatial-temporal domain Recent research has started investigating completely new
n fromjustafewcarefully-chosensamples.Here,wepresentanew
techniques for multi-way classification that attempt to better
a method,referredtoastheSparse-TDAalgorithm,thatcombines
J favorable aspects of the two techniques. This combination is understand the structure of the underlying high-dimensional
2 realized by selecting an optimal set of sparse pixel samples sample space. One such class of techniques is topological
1 from the persistent features generated by a vector-based TDA data analysis, or TDA in short. TDA represents the unknown
algorithm. These sparse samples are selected using a low-rank sample space in the form of persistent shape descriptors

] matrix representation of persistent features. We show that the that are coordinate free and deformation invariant. Thus,
L Sparse-TDA method demonstrates promising performance on
the descriptors define topological features and yield insights
M threebenchmarkproblemsrelatedtohumanposturerecognition
and image texture classification. regarding suitable feature selection.
. Another critical tool facilitating multi-way classification
t Index Terms—Topological data analysis, sparse sampling,
a
is the feature-driven sparse sampling of high-dimensional
t multi-wayclassification,humanposturedata,imagetexturedata.
s data. Observations are typically sparse in a transform basis
[
of informative features, so measurements can be optimally

1 I. INTRODUCTION chosen to enhance discriminating features in the data. This
2v MULTI-way or multi-class classification, where the goal permits heavily subsampled inputs for downstream classifiers,
which drastically reduces the burdens of sample acquisition,
1 is to correctly predict one out of K classes for any
processing and storage without sacrificing performance.
2 data sample, poses one of the most challenging problems
3 Here, we bring together the two research areas of TDA and
in supervised machine learning. However, a large number of
0 sparse sampling in the context of multi-way classification. In
real-world sensing problems in a variety of domains such as
1. computer vision, robotics and remote diagnostics, do consist particular, we leverage sparse sampling for optimal feature
0 selection once the features are extracted using a state-of-the-
of multiple classes. Examples include human face recognition
7 artTDAmethodinchallengingcomputervisionproblems.The
for surveillance, object detection for mobile robot navigation,
1
problems comprise three benchmark data sets pertaining to
: and critical equipment condition monitoring for preventive
v 3D meshes of synthetic and real human postures and textured
maintenance. The number of classes in these problems often
i images,respectively.WecallournewmethodtheSparse-TDA
X exceeds ten and sometimes goes up to a hundred depending
algorithm and show that it achieves comparable accuracy as
r on the complexity of the sensed system or environment and
a theTDAmethodwithsignificantlylowertrainingtimes.Thus,
the number and types of sensor modalities.
our method opens up a new direction in making online multi-
While a whole host of techniques such as artificial neural
way classification practically feasible.
networks, decision trees, na¨ıve Bayes, nearest neighbors, and
support vector machines (SVMs) have been successfully ap-
plied for binary classification problems, extensions of these II. RELATEDWORK
techniques have had mixed success in addressing multi-way
Overthepastdecadeorso,anincreasinginterestinutilizing
classificationproblemswithmorethanafewclasses.Otherap-
tools from algebraic topology to extract insights from high
proachesinvolvinghierarchicalclassificationortransformation
dimensional data has given rise to the field of TDA. The
W. Guo is with the Department of Industrial & Systems Engineering, successful applications of TDA have spanned a large number
UniversityofWashington,Seattle,WA,98195USAe-mail:[email protected]. ofareas,rangingfromcomputervision[1]tomedicalimaging
K.ManohariswiththeDepartmentofAppliedMathematics,Universityof
[2], biochemistry [3], neuroscience [4] and materials science
Washington,Seattle,WA,98195USAe-mail:[email protected].
S.L.BruntoniswiththeDepartmentofMechanicalEngineering,University [5].ApredominanttoolinTDAispersistenthomology,which
ofWashington,Seattle,WA,98195USAe-mail:[email protected]. tracks the evolution of the topological features in a multi-
A.G.BanerjeeiswiththeDepartmentofIndustrial&SystemsEngineer-
scale manner to avoid information loss [6], [7]. The multi-
ing and Department of Mechanical Engineering, University of Washington,
Seattle,WA,98195USAe-mail:[email protected]. scale information is summarized by the persistence diagram
IEEETRANSACTIONSONKNOWLEDGEANDDATAENGINEERING 2
(PD),amultisetofpointsinR2 thatencodesthelifetime(i.e., simplicial (or cubical) complex S is a sequence of simplicial
persistence) of the features. complexes S ,...,S such that S ⊆ ... ⊆ S = S. A
1 r 1 r
More recently, researchers have started utilizing TDA for commonwaytogenerateafiltrationistoconsiderthesublevel
machine learning problems. Pachauri et al. [8] first computed sets S = f−1([−∞,t]) of a descriptor function f : X → R
t
aGaussiankerneltoestimatethedensityofpointsonaregular on a topological space X indexed by a parameter t∈R.
grid for each rasterized PD, and fed the discrete density esti- Persistenthomology:AsaprevalenttoolinTDA,persistent
mation as a vector into an SVM classifier without any feature homology is an algebraic approach that quantifies topological
selection.However,theirmethoddidnotestablishthestability features during a filtration of “shapes” [6], [7]. Accordingly,
ofthekernel-inducedvectorrepresentation.Reininghausetal. givenatopologicalspaceXandadescriptorfunctionf :X→
[1]thendesignedastablemulti-scalekernelforPDsmotivated R, persistent homology essentially studies the topological
byscale-spacetheoryaswillbedescribedinthenextSection. changesofthesublevelsetsX =f−1([−∞,t])astincreases
t
Experiments on three benchmark data sets showed that this from −∞ to ∞. During filtration, topological features appear
method greatly outperformed an alternative approach based and disappear at different scales that are referred to as the
on persistence landscape [9], a popular statistical treatment birth and death times of the features. The short-lived features
of TDA. Similar to this work, Kusano et al. [10] proposed a are considered as noise terms.
stable persistence weighted Gaussian kernel, allowing one to Persistence diagram (PD): A PD is a concise summary of
control the effect of persistence. However, the computational the topological information captured by persistent homology.
complexity of both the kernel-based methods for calculating From a geometric perspective, the topological features are
the Gram matrix is O(m2n2) if there are n PDs for training interpretedasl-dimensionalholes,e.g.,connectedcomponents
and the PDs contain at most m points, which can be quite as 0-dimension holes, tunnels as 1-dimensional holes and
expensive for many practical applications. voids as 2-dimensional holes. Thus, a l-dimensional PD is a
To enable large-scale computations with PDs, recent meth- collection of points in R2, where each point (a,b) represents
odshavemappedeachPDtoastablevectortoallowdirectuse al-dimensionalholethatisbornattimeaandfilledattimeb.
of vector-based learning methods. For example, Adams et al. Stability: A critical property of PDs is their stability with
[11] constructed vectors by discretizing the weighted sum of respect to input noise [20], [21]. A general metric associated
probability distributions centered at each point in transformed with PDs is the p-Wasserstein distance. The p-Wasserstein
PDs. Carrie`re et al. [12] rearranged the entries of the distance distance between two PDs F and G is defined by
matrix between points in a PD and Bonis et al. [13] adopted
(cid:18) (cid:19)1
a pooling scheme to construct the vectors. d (F,G)= inf (cid:88)||x−γ(x)||p p, 1≤p≤∞, (1)
W,p ∞
Existing sparse sampling techniques leverage empirical in- γ
x∈F
terpolationmethods(EIM)[14]–[16]toidentifydiscriminating
where γ : F → G is over all bijections from the points in F
measurements using low-rank spanning features, such as ob-
to the points in G.
tained by principal components analysis (PCA). This strategy
Let X be a compact triangulable metric space and f,g :
has been employed for sparse classification of dynamical
X →R be two tame Lipschitz functions with the correspond-
regimeswithinlibrariesoflearnedPCAfeatures[17],[18].In
ing PDs D (f) and D (g) for each dimension l. It has been
image classification using linear discriminant analysis (LDA), l l
prove that assuming X satisfies a weak condition (see details
optimized sparse pixel locations were determined that map
in [21]), there exist constants q ≥1 and C, which depend on
into the discriminating subspace in PCA coordinates [19].
X and the Lipschitz constants of f and g, such that
Inthiswork,weemploythevectorrepresentationfrom[11]
and integrate with a sparse sampling method using QR pivots 1−q
d (D (f),D (g))≤C||f −g|| p, p≥q. (2)
W,p l l ∞
toidentifydiscriminativefeaturesinthepresenceofnoisyand
redundant information to further improve classifier training This upper bound on d (D (f),D (g)) implies that a PD
W,p l l
time and sometimes prediction accuracy. D (f) is stable with respect to the p-Wasserstein distance
l
under small perturbations of f.
III. KERNEL-BASEDTDAMETHODFORMULTI-WAY Kernel: Given a set X, k : X × X → R is a kernel if
CLASSIFICATION there exists a Hilbert space H, called feature space, and a
map Φ:X →H such that
WefirstintroducethebasicterminologyofanyTDAmethod
by formally defining the mathematical operations and stating k(x,x(cid:48))=(cid:104)Φ(x),Φ(x(cid:48))(cid:105) (3)
H
the key results. We then use these definitions and results to
summarize the multi-scale kernel TDA method. While the for all x,x(cid:48) ∈ X. In machine learning, a kernel represents
kernel TDA method is directly adopted from [1], we outline a similarity measure between the samples, and Φ is called its
it here for the sake of completeness. featuremap.AkernelsatisfyingEq.(3)isalsosymmetricand
positive definite [22].
A. Terminology
B. Multi-scale Kernel TDA
Filtration: In computational topology, the “shapes” of data
(e.g., point clouds, shapes, images) are frequently described In[1],Reininghausetal.devisethepersistencescalespace
by simplicial (or cubical) complexes. A filtration of a finite kernelonthesetofPDD asamulti-scalekernelviaafeature
IEEETRANSACTIONSONKNOWLEDGEANDDATAENGINEERING 3
map Φ : D → L (Ω), where Ω = {x = (x ,x ) ∈ R2 : A persistence surface ρ :R2 →R on T(D ) is defined by
σ 2 1 2 Di i
x ≥ x } denotes the space above the diagonal. Given a PD (cid:88)
2 1 ρ (z)= f(u)g (z) (9)
D ∈D, the feature map Φ is the solution of a heat diffusion Di u
σ
problem with a Dirichlet boundary condition on the diagonal: u∈T(Di)
where f : R2 → R is a non-negative weighting function that
Φσ(D)= 4π1σ (cid:88)(cid:16)e−||x−4σy||2 −e−||x−4σy¯||2(cid:17), (4) idsiffzeerreontailaobnleg;tghe h:oRri2zo→ntalRaxiiss,acpornotibnauboiluisty, afnudncptiioencewwiisthe
y∈D u
mean u=(u ,u )∈R2 and variance σ2.
x y
where y¯=(b,a) is the mirror image of y =(a,b) across the
In our experiments, the linear weighting function is
diagonal. The map then yields the kernel k :D×D →R in
σ u
a closed form as f(u)= y, (10)
u∗
y
k (F,G)=(cid:104)Φ (F),Φ (G)(cid:105)
σ = 1σ (cid:88)(cid:16)eσ−||y−8σLz|2|2(Ω−) e−||y−8σz¯||2(cid:17) (5) wlinheearrewue∗yig=htimngaxfui=nc1,t.i.o.,nnimsainxsup∈irDeidubyy.tThheewfeoirgmhtionfgtfhuencntoionn-
8πσ used in [10] and chosen as
y∈F
z∈G
f(u)=arctan(cu ). (11)
y
for σ > 0 and F,G ∈ D, which has been shown to be 1-
where c=(median median u )−1. We choose g
Wasserstein stable. Further, note that because the summation i=1,...,n u∈Di y u
to be the Gaussian distribution, i.e.,
in Eq. (5) is carried out over all pairwise combinations of the
1
points in the PDs F and G, evaluation of the kernel requires g (z)= e−[(zx−ux)2+(zy−uy)2]/2σ2. (12)
O(|F||G|) time, where |F| and |G| denote the number of u 2πσ2
points in F and G, respectively. where z = (zx,zy). Then the PI, a matrix of pixel values, is
obtained by calculating the integral of ρ on each grid box
Di
from discretization,
IV. SPARSE-TDAMETHOD (cid:90)(cid:90)
We now introduce a vector representation of a PD, termed I(ρDi)= ρDi(zx,zy)dzxdzy. (13)
a persistence image (PI), presented in [11]. Since our Sparse-
PI has also been proven to be 1-Wasserstein stable. Let the
TDA method will combine PI-based TDA with sparse sample
grid resolution for each PI be r×r. We reshape each PI to be
selection, we first summarize the sparse sampling method a vector x ∈Rr2, i=1,...,n. Our sparse sample selection
before describing the combination. i
methodisunsupervised,wherewethenstackthevectorsfrom
all the classes columnwise into a matrix as
A. Optimized Sparse Sample Selection X∈Rr2×n =[x | x |...| x ]. (14)
1 2 n
Samples are selected from low-rank feature vectors using
Assume that the number of desired features is s. Applying
the pivoted QR matrix factorization, also known as Q-DEIM
the sparse sampling method on X, we obtain the row index
[16]. Assuming observations are columns in a data matrix X,
of s optimal feature locations and the reduced feature vectors
wefirstobtainasparsifyingbasisforitscolumnsfromtheleft x˜ ,x˜ ,...,x˜ ∈Rs for the corresponding classifiers.
1 2 n
singular vectors of its singular value decomposition
V. RESULTS
X≈U Σ VT, (6)
s s s WenowdiscusstheperformanceofourSparse-TDAmethod
where s is the rank of the truncated basis and the number of on three benchmark computer vision data sets. The data sets
desired measurements. The column pivoted QR factorization are explained first, followed by illustrations of the selected
of UT optimizes a row permutation of the basis that is features and quantitative comparisons of our method with
s
numerically well-conditioned by expressing UT as a product the multi-scale kernel TDA method. The illustrations and
s
of unitary and upper-triangular matrices Q and R comparison results show the usefulness of the method on
challenging multi-way classification problems.
UTΠT =QR. (7)
s
A. Data Sets
The optimized row permutation is ΠU , and the sparse
s For shape classification, SHREC’14 synthetic and real data
measurements of an observation x are given by x˜ =Πx.
sets are used, given in the format of triangulated 3D meshes
[23]. The synthetic set contains meshes from five males, five
B. Combining Sparse Feature Selection with Persistence Im- females and five children in 20 different poses, while the real
ages set consists of 20 males and 20 females in 10 different poses.
For texture recognition, we use the Outex_TC_00000
Let D = {D | i = 1,...,n} be a training set of PDs. To
i
data set [24]. This data set contains 480 images equally
constructaPIfromagivenPDD [11],D isfirsttransformed
i i
categorizedinto24classesandprovides100predefined50/50
from birth-death coordinates to birth-persistence coordinates.
Let T :R2 →R2 be the linear transformation, training/testing splits. During preprocessing, we downsample
the original images to 32×32 pixel images as done in the
T(x,y)=(x,y−x). (8) multi-scale kernel TDA method.
IEEETRANSACTIONSONKNOWLEDGEANDDATAENGINEERING 4
B. Feature Selection set. Results are reported based on 30 runs for each case with
the exception of those presented for the OuTex data set in
We first follow the same procedure performed in the multi-
Tables I-III, which are based on 100 runs.
scale kernel TDA method to obtain PDs. For SHREC’14 data
Table II compares the classification accuracy of both the
sets, we compute the heat kernel signature [25] on the surface
variantsofourSparse-TDAmethodwiththemulti-scalekernel
mesh of each object and then compute the 1-dimensional
PDs using Dipha1. For the OuTeX data set, we take the sign TDA method. Note that the two SHREC’14 data sets are
partitioned into 70/30 training/testing samples, whereas the
componentofthecompletedlocalbinarypatternoperator[26]
OuTeX set is partitioned into 50/50 training/testing samples.
asthedescriptorfunction.Thenwegeneratethe0-dimensional
The number of samples for each class is approximately the
PDs from the filtration of its rotation-invariant version with
same in all the training sets. Overall, the accuracy results are
P =8 neighbors and radius R=1.
comparable for all the three benchmark data sets. However,
TogeneratethePIs,wesettheresolutiontobe30×30for
consistent with the differences observed in the PIs among
all three data sets. In fact, the classification accuracy is fairly
the classes, both our linear and non-linear weighting methods
robusttothechoiceofresolution[11].Wealsosetσtobe0.15,
perform slightly better than the kernel TDA method for the
0.0001 and 0.02 for SHREC’14 synthetic, SHREC’14 real
SHREC’14 real data set. On the other hand, our method is
and OuTex data sets, respectively. Fig. 1 shows representative
marginally worse than the kernel TDA method for both the
PIs for three different classes in all of our benchmark data
SHREC’14 synthetic and OuTeX data sets, even though the
sets. Noticeable differences are observed among the PIs for
accuracy increases slightly using non-linear weighting.
each of the three data sets, although the differences are most
pronounced for the SHREC’14 synthetic data set, reasonably
clear for the SHREC’14 synthetic data set, and less evident TABLEII
COMPARISONOFCLASSIFICATIONACCURACY(IN%)WITHTHESAME
for the OuTeX data set. These differences in the pixel values
TRAININGANDTESTDATASETSPLITASINTHEORIGINALMULTI-SCALE
of the PIs form the distinguishing class features from which KERNELTDAARTICLE
an optimal set is selected by QR pivots.
Sparse-TDA
DataSet Linear Non-linear KernelTDA
TABLEI
weighting weighting
SIZEOFTHETRAININGSETSANDENERGYCONTAINEDINTHE100
SHREC’14Synthetic 91.7±2.6 93.6±2.7 98.4±1.7
OPTIMIZEDPIXELLOCATIONS(IN%)
SHREC’14Real 67.3±4.1 67.9±4.6 65.9±4.4
Sparse-TDA OuTeXTexture 62.5±2.9 66.0±2.5 69.2±2.4
DataSet TrainingSize Linear Non-linear
weighting weighting Table III provides a comparison of the three methods in
SHREC’14Synthetic 210 100±0 100±0 terms of the kernel SVM-based classifier training times that
SHREC’14Real 280 97.8±0.13 97.6±0.12 aremeasuredonalaptopwithanIntelCorei5(2.4GHz)CPU
OuTeXTexture 240 89.5±0.12 91.0±0.12 and4GBRAM.IntheSparse-TDAmethod,thetrainingtimes
start from the computation of PIs. Not surprisingly, both of
Pixels are selected using the QR pivots of the first s=100 our method variants are usually much faster than kernel TDA
left singular vectors in each dataset, therefore Sparse-TDA as they use smaller sets of selected features. As expected, the
accuracy is affected by how much energy or variance in each reductionintrainingtimeisgreaterwithlinearweightingthan
training set is explained within the first 100 basis vectors with non-linear weighting. In fact, the Sparse-TDA method
Us. The energy of each training set, tabulated in Table I, is with linear weighting achieves about 65X speed-up for the
the normalized sum of the first 100 singular values (diagonal SHREC’14 synthetic data set and roughly 8X speed-up for
entries of Σ). For instance, it can be seen that the SHREC’14 the OuTeX data set. However, there is no consistent speed-up
synthetic data set is intrinsically lower rank. fortheSHREC’14realdatasetowingtothefactthatthereare
only 4-5 points in each PD in this case, rendering the training
of the kernel TDA method exceptionally fast.
C. Classification Performance
We feed the reduced feature vectors for training into a
TABLEIII
soft margin C-SVM classifier with a radial basis function COMPARISONOFCLASSIFIERTRAININGTIMES(INS)WITHTHESAME
(RBF) kernel, implemented in LIBSVM [27], for each data TRAININGANDTESTDATASETSPLITASINTHEORIGINALMULTI-SCALE
set. The cost factor C and kernel parameter γ are tuned KERNELTDAARTICLE
based on a grid search using 10-fold cross-validation on the Sparse-TDA
training data. We start a coarse grid search with exponentially DataSet Linear Non-linear KernelTDA
growing sequences of C and γ first, thereafter proceeding weighting weighting
with finer grid searches in the vicinity of the optimal region SHREC’14Synthetic 18.2±0.95 43.6±7.09 1220±27.9
yieldedbythepreviousgridsearch.Eachgridsearchincludes SHREC’14Real 55.4±1.1 160±54.4 107.8±5.4
a total of 50 pairs of (C,γ) values which are used to apply OuTeXTexture 693±82.4 834±127 5457±979
the training model to the reduced feature vectors of the test
Fig.2andFig.3showthetrendsinimprovingtheclassifica-
1https://github.com/DIPHA/dipha tionaccuracyandreducingtheclassifiertrainingtime,respec-
IEEETRANSACTIONSONKNOWLEDGEANDDATAENGINEERING 5
(a)SHREC’14syntheticpostures (b)SHREC’14realpostures (c)OuTeXtextureimages
(d)SHREC’14syntheticPIsusinglinearweight- (e)SHREC’14realPIsusinglinearweighting (f)OuTeXtexturePIsusinglinearweighting
ing
(g) SHREC’14 synthetic PIs using non-linear (h)SHREC’14realPIsusingnon-linearweighting (i)OuTeXtexturePIsusingnon-linearweighting
weighting
Fig.1. Representativeclassesandcorrespondingpersistenceimages(PIs)forvariousbenchmarkdatasetsusingtwoSparse-TDAmethods.
tively, as a function of increasing training/testing split for all image texture detection. While our method gives comparable
the benchmark data sets. Consistent with the results reported classification accuracy to the multi-scale kernel method, it
in Tables II, our classification accuracy is marginally inferior resultsinasignificantreductionoftheclassifiertrainingtimes.
tothatofthekernelTDAmethodfortheSHREC’14synthetic This reduction is expected to lay the foundation for on-line
and OuTeX data sets. However, the Sparse-TDA method with adaptation of TDA on challenging data sets, which have a
linear and non-linear weighting marginally outperform the large number of classes, in response to changes in training
kernel method for the most challenging SHREC’14 real data sample availability.
set.Thetrainingtimetrendsarealsoverysimilartotheresults Inthefuture,wewouldliketofurtherimprovetheaccuracy
presented earlier in Table III, with more than an order of of the Sparse-TDA method by designing our own weighting
magnitude reduction for the SHREC’14 synthetic data set, function for the persistence images. We would also like to
comparable values for the SHREC’14 real set, and significant come up with theoretical performance guarantees based on
reduction for the OuTeX data set. The increase in classifier the characteristics of the data sets, particularly the training
training times with higher training/test splits is, however, sample size for each individual class. Last but not the least,
slightlymoreforboththevariantsofourmethodascompared weplantoshowtheeffectivenessofourmethodonotherhard
to the kernel TDA method. classification problems arising in robot visual perception and
Overall, we observe that, for each of the benchmark data human face recognition.
sets, at least one of the variants of our Sparse-TDA method
outperformsthemulti-scalekernelTDAmethodeitherinterms
ACKNOWLEDGMENTS
of classification accuracy or classifier training time.
We would like to thank The Boeing Company for sponsor-
ing this work in part under contract # SSOW-BRT-W0714-
VI. CONCLUSIONS
0004 and Dr. Tom Hogan for helpful discussions. The views
In this paper, we present a new method, referred to as the
and opinions expressed in the paper are, however, solely of
Sparse-TDA algorithm, that provides a sparse realization of a
theauthorsanddonotnecessarilyreflectthoseofthesponsor.
topological data analysis (TDA) algorithm. More specifically,
we combine an optimal sparse sample selection method based
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