3.5 Comparison Results of Nonlinear Dimensionality Reduction ...

3.5 Comparison Results of Nonlinear Dimensionality Reduction Techniques 25

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Figure 3-6.: Nonlinear Feature Extraction the Swiss roll with hole database

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Figure 3-7.: Nonlinear Feature Extraction for the fishbowl database

26 3 Comparative Analysis of Nonlinear Feature Extraction Techniques

3-3 and Table 3-4, present the quality of the embedding for the real-world datasets. On the

other hand, Table 3-5 and 3-6 show the computational time for all the methods for 2-D and

3-D representations.


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Figure 3-8.: 2-D representation for the Manki Neko database


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Table 3-1.: PNE for synthetic datasets, ideal PNE= 0

Method Database

Fishbowl S-Surface Swiss Role with Hole

ISOMAP 0, 4393 0, 3660 0, 5135

LLE 0, 0062 0,0009 0, 1020

LEM 0,0032 0, 0052 0, 238

MVU 0, 0121 0,0009 0,0031

Table 3-2.: Computational load in seconds. 2-D representations of the synthetic datasets.

Method Database

Fishbowl S-Surface Swiss Role with Hole

ISOMAP 52, 38 53, 05 49, 74

LLE 24, 25 23, 67 26, 30

LEM 81, 57 83, 11 33, 35

MVU 13.8987 8.178, 2 9.592, 4

PCA 0,07 0,01 0,01

t-SNE 83, 91 87, 01 96, 13

Table 3-3.: PNE for real-world datasets – 2-D representations, ideal PNE = 0

Method Database

Maneki Neko Frog Duck

ISOMAP 0, 0427 0, 0686 0, 0575

LLE 0, 0444 0, 0404 0,0221

LEM 0,0184 0,0167 0, 0334

MVU 0, 0492 0, 0538 0, 0342

Table 3-4.: PNE for real-world datasets – 3-D representations, ideal PNE = 0

Method Database


ISOMAP 0, 2085 0, 1939 0, 2436

LLE 0, 0418 0, 0418 0,0205

LEM 0,0220 0,0276 0, 0483

MVU 0, 0454 0, 0451 0, 0310


Table 3-5.: Computational load in seconds. 2-D representations of the real-world datasets.

Method Database


ISOMAP 1, 90 1.95 1, 96

LLE 2, 25 2, 27 2, 34

LEM 3, 05 2, 59 2, 23

MVU 3, 39 3, 74 3, 88

PCA 0,03 0,31 0,04

t-SNE 1, 02 1, 38 0, 95

Table 3-6.: Computational load in seconds. 3-D representations of the real-world datasets.

Method Database


ISOMAP 1, 90 1, 93 1, 87

LLE 2, 23 3, 10 2, 36

LEM 2, 33 2, 42 2, 37

MVU 3, 30 3, 69 3, 87

PCA 0,03 0,03 0,03

t-SNE 1, 08 1, 00 0, 97

3.6 Discussion 35

3.6. Discussion

The conventional PCA does not achieve adequate results for the synthetic datasets, due to

the low-dimensional representations exhibit overlapping, as can be seen in Figures 3.7(a),

3.5(a) and 3.6(a). That is, PCA does not seek to unfold the input data, but it finds the

projection that preserves the variance of data as much as possible without taking into account

neither the local nor global data structure. Similar results are obtained for the real-world

datasets, where the behavior of the activity is lost in the mapping process(Figures 3.8(a),

3.9(a), 3.10(a), 3.11(a), 3.12(a) and 3.13(a)). In this regard, PCA is not appropriate for

analyzing data that lie in nonlinear manifold, specially for visualization tasks.

In contrast, other methods like ISOMAP, LLE, LEM and MVU find low-dimensional

spaces that suitably represent the data. However, the performance of these techniques is

not constant for all the databases. For instance, ISOMAP is not able to properly unfold the

Fishbowl dataset (see Figure 3.7(b)), but shows better results for the S-Surface and the Swiss

Role with Hole, getting faithful representations of the high dimensional data (Figures 3.5(b)

and 3.6(b)). It is possible to observe that in these cases some holes or blank patches appear

in the output space, which is one of the weakness of ISOMAP that is common when data is

not uniform and well sampled. For the COIL-100 database, the 2-D embeddings obtained

with ISOMAP exhibit a symmetrical and consistent global structure (Figures 5.11(a), 3.9(b)

and 3.10(b)), specially for the Maneki Neko and Duck datasets. For The 3-D embeddings,

ISOMAP tries to preserve the global data structure, but the results are not as smooth as

the other tested methods (Figures 3.11(b), 3.12(b) and 3.13(b)).

LLE demonstrates a good performance dealing with synthetic data. Nevertheless, there

is a great influence of the regularization parameter (when k > p) in low-dimensional repre-

sentations. According to the results (Figures 3.7(c), 3.5(c) and 3.6(c)), this method seeks

to unfold input data preserving both global and local structure, but most of the times it

produces a change of scale or rotation in the low-dimensional space, as in the Swiss Role

with Hole and S-Surface datasets. Table 3-1 shows that the quality of LLE embeddings are

not the lower ones, and some times it gets the best transformation. Again, for the real-world

datasets, LLE finds suitable low-dimensional representations that clearly allow to identify

the object rotation, which is due to the preservation of local properties of the input data,

as shown in Figures 3.8(c), 3.9(c) and 3.10(c). In this sense, the results are quite similar to

those found with other techniques such as LEM and ISOMAP for the 2-D representations.

Now, for the 3-D embeddings the performance of LLE is higher than ISOMAP according to

Table 3-4, which can be visually confirmed in Figures 3.11(c), 3.12(c) and 3.13(c).

The LEM method has difficulties to compute appropriate transformations for the artificial

datasets. Even when it tries to unfold data in order to conserve the global properties, the

local relationships are not fully preserved, which generates two possible consequences: blank

patches or overlapping in the low-dimensional space. These characteristics are evident in

Figures 3.7(d), 3.5(d) and 3.6(d). This may be explain by considering that LEM attempts


to be insensitive to outliers, and for that reason if the manifold is not well sample or is

non-uniform it adds a penalty and the local geometry is not properly conserve. Nonetheless,

LEM has the higher performance for the Fishbowl dataset, which is one of the most difficult

to unfold considering its intrinsic geometry. In general, it is possible to say that LEM

is not the best option to handle synthetic datasets for visualization purposes. Otherwise,

the embeddings obtained for the real-world databases are much better, and LEM shows the

higher performance in two of the three tested databases for both 2-D and 3-D representations

(see Table 3-3 and 3-4). In this sense, the LEM technique proves to be a good option to

work with real-word data, finding embeddings that preserve the local and global structure.

Moreover, it has a good performance for both 2-D and 3-D visualization tasks (Figures

3.8(d), 3.9(d), 3.10(d), 3.11(d), 3.12(d) and 3.13(d)).

In regard to the synthetic datasets, MVU presents the higher performance for the S-surface

and the Swiss Roll with Hole, as can be confirm in Table 3-1. This technique seeks to find a

transformation that preserves both global and local data structure simultaneously, which is

implicit in the constrains of the optimization problem. As a consequence, data is properly

unfolded in the low-dimensional space, exhibiting symmetry (see Figures 3.5(e) and 3.6(e)).

Specifically, for the Fishbowl, MVU tries to unfold the data points but the output space

shows overlapping 3.7(e). It is important to note that this manifold has regions with low

density of samples, which affects the performance of the algorithms. MVU’s weakness lies

in the fact that the SDP optimization problem requires a high computational load, and for

that reason while other techniques only need a few minutes to perform the analysis, MVU

demands hours, see Table 3-2. This time demand decreases when the amount of samples is

lower, and therefore the computational load for the tested real-world datasets is acceptable

and competitive with the other techniques (see Table 3-5 and 3-6). However, the quality of

the embedding results is not as good for the real-world datasets. In this cases, the results still

show symmetry, but the local data structure is not well preserved. So, even if the rotation

behavior can be infer from the low-dimensional representation, the local properties are not

present in the output space (Figures 3.8(e), 3.9(e), 3.10(e), 3.11(e), 3.12(e) and 3.13(e)).

Finally, with the purpose of comparing the manifold learning methods with another kind

of techniques for dimensionality reduction, the t-SNE algorithm was taken into account in

this analysis. Unlike the above techniques, t-SNE is based on probabilistic theory, which add

another type of parameters to the transformation related to the chosen distribution. With

respect to the artificial databases the low-dimensional representations found with t-SNE

have a common characteristic, they present a considerable amount of blank patches, thence

the data seems to be disconnected, and the local properties of the input data are not fully

conserved,, as can be notice in Figures 3.7(f), 3.5(f) and 3.6(f). Another weakness about the

t-SNE method is to tune the initial condition of the algorithm, which can change every time

that the method is used, affecting the transformation, and hence, the output is quite different

every time. For the real-world datasets the results are similar, the embeddings do not reflect

the rotation behavior, and the local structure of data is not preserved by neither the 2-D

3.6 Discussion 37

nor 3-D representations (see Figures 3.8(f), 3.9(f), 3.10(f), 3.11(f), 3.12(f) and 3.13(f)).

4. Multiple Manifold Learning for

Dimensionality Reduction

Often, in machine learning and pattern recognition literature, the nonlinear feature extrac-

tion (NFE) techniques are reviewed as learning methods for discovering an underlying low-

dimensional structure from a set of high-dimensional input samples, that is, NFE techniques

unfold a non-linear manifold embedded within a higher-dimensional space. Nevertheless,

most of the NFE algorithms are constrained to deal with a single manifold, attaining unap-

propriate low-dimensional representations when input data lie on multiple manifold, because

the inter-manifold distance is usually much larger than the intra-manifold distance [33], mov-

ing apart each manifold from the others, regardless of whether the behavior among them is

similar.

To our best knowledge, some few works [23, 33, 34] have been proposed in order to allow

the application of the NFE techniques on multiple manifold data sets. Particularly, in [33]

a framework to learn an embedded manifold representation from multiple data sets called

Learning a Joint Manifold (LJM) is presented, which finds a common manifold among the

different data sets, without assuming some kind of correspondence between the different

manifolds. However, the main drawback of this approach is that for obtaining suitable

low dimensional representations the input samples must be similar in appearance, when

the multiple manifolds do not have a close resemblance among them the LJM method fails

to embed the data. On the other hand, the approach presented in [34] actually uses a

correspondence labels among the samples in order align the data sets, in such case the

complexity of the challenge is lower than when no one correspondence is assumed. A similar

solution is proposed in [35].

Unlike these mentioned works for dealing with multiple manifolds, our major contribution

to the state of the art is the possibility for analyzing dissimilar objects/subjects in appearance

but with a common behavior (similar motion), moreover our methodology allows to employ

objects/subjects with different input dimensions and number of samples among manifolds.

Our approach is inspired on the manifold learning algorithm Laplacian Eigenmaps - LEM

[18](described in Section 3.1.5), because its optimization problem has an analytic solution

avoiding local minima, and few free parameters need to be fixed by user. Our approach can

be employed to visually identify in a low-dimensional space the dynamics of a given activity

learning it from a variety of datasets. We test on two real-world databases, changing the

number of samples and input dimensions per manifold. Our proposal is compared against

4.1 Multiple Manifold Learning - MML 39

both the conventional Laplacian Eigenmaps (LEM) [23] and the closest work found in the

state of the art for multiple-manifold learning (LJM) [33]. Overall, our methodology achieves

meaningful low dimensional representations, visually outperforming the results obtained by

the other methods.

Next section (4.1) introduces the proposed methodology for multiple manifold dimension-

ality reduction. Then, the experimental conditions are presented, and finally the results are

described and discussed.

4.1. Multiple Manifold Learning - MML

The NFE frameworks based on manifold learning such as ISOMAP, LLE and LEM were

shown to be able to embed data lying on a nonlinear manifold into a low-dimensional Eu-

clidean space for synthetic datasets, as well as for real images (for example the COIL-100

Database tested in Chapter 3). Nonetheless, these techniques fail when they look for a com-

mon low-dimensional representation for data lying on multiple manifolds or multiple classes.

Considering that in real world applications, like video and image analysis, datasets with mul-

tiple manifolds are common, we propose a framework to find a low-dimensional embedding

for data lying on multiple manifolds. The proposed approach is inspired on the manifold

learning algorithm Laplacian Eigenmaps - LEM, which was chosen due to the mathematical

formulation allows to add information to the mapping process. The main goal is to compute

the relationships among samples of different datasets based on an intra manifold compari-

son to unfold properly the data underlying structure taking into account that we are given

multiple data sets and there is an underlying common manifold among them.

In this sense, we propose to related each input sample xi with C different manifolds

that share a similar underlying structure. Let ΨΨΨ = {Xc}Cc=1 an input manifold set, where

Xc ∈ Rnc×pc . Our goal is to find a mapping from ΨΨΨ to a low-dimensional space Y ∈ Rn×m

(with m� pc, and n =∑C

c=1 nc), which reveals both the intra manifold structure (relation-

ships within manifold), and the inter manifold structure (relationships among manifolds).

Consequently, a weight matrix A that takes into account both structures can be computed

as

A =

W1 M12 · · · M1c · · · M1C

M21 W2 · · · M2c · · · M2C

......

. . ....

. . ....

Mc1 Mc2 · · · Wc · · · McC

......

. . ....

. . ....

MC1 MC2 · · · MCc · · · WC

, (4-1)

where each Wc ∈ Rnc×nc is the traditional LEM intra manifold weight matrix for each Xc

[33]. Furthermore, each Mcb ∈ Rnc×nb (b = 1, . . . , C) block is a soft correspondence matrix

40 4 Multiple Manifold Learning for Dimensionality Reduction

between Xc and Xb.

In a previous work [33], is proposed a methodology called Learning a Joint Manifold

Representation (LJM) to unfold the data underlying structure from multiple manifolds. LJM

calculates the matrix A (equation (4-1)), computing the intra manifold structure matrices

Wc as in traditional LEM, and the inter manifold structure matrices Mcb by solving a

permutation matrix P, which allows to find a maximum weight matching by permuting the

rows of Ucb ∈ Rnc×nb , U cbqr = κ (xq,xr), xq ∈ Xc, and xr ∈ Xb (q = 1, . . . , nc; r = 1, . . . , nb ).

Nonetheless, LJM is sensitive to feature variability between samples of different manifolds,

due to Ucb is inferred in the high-dimensional space. Moreover, LJM is limited to analyze

input matrices Xc which belong to the same input dimension (p1 = p2 = · · · = pc = · · · =

pC), as can be seen in the calculation of each Ucb.

In this work, we identify the correspondence among data points from different manifolds

without making a high-dimensional sample comparison. In other words, the similarities

among observations of different manifolds are not directly calculated in each pair Xc and

Xb. Therefore, we compute each soft correspondence matrix Mcb in (4-1) as

M cbqr =

⟨wcq,w

br

⟩∥∥wcq

∥∥ ‖wbr‖, (4-2)

where wcq ∈ R1×nc and wb

r ∈ R1×nb are row vectors of Wc and Wb, respectively. It is

important to note that equation (4-2) is not well defined when nc 6= nb, thereby, we use

an interpolation method based on cubic splines for oversampling the lowest size vector, to

properly compute the inner product between wcq and wb

r. Our approach for Multiple Manifold

Learning (MML) aims to calculate the relationships among samples of different manifolds,

comparing the intra manifold similarities contained in each Wc (equation (4-2)). Finally,

given the weight matrix A, we minimize the following objective function∑ij

(yi − yj)2Aij. (4-3)

Solving equation (4-3) as in traditional LEM algorithm allows us to find a low-dimensional

space Y for data lying on multiple manifolds.

4.2. Experiments and Results

We tested the conventional LEM algorithm, the state of the art methodology named LJM,

and our proposed methodology called MML on two real-world databases, in order to find a

2-D low-dimensional representation (m = 2) for data lying on multiple manifolds.

The first database, the Columbia Object Image Library (COIL-100) [36], contains 72

RGB-color images, for each one of the 100 objects, in PNG format, which were taken at pose

intervals of 5 degrees while the object is rotated 360 degrees. In this work, the following

4.2 Experiments and Results 41

objects are used: Car, Frog and Duck. The image size is 128× 128, which are transformed

to gray scale. In Figure 3-4 are shown some examples.

The second database is the CMU motion of body (Mobo)[37], which holds 25 individuals

walking a treadmill in the CMU 3-D room. The subjects perform four different walk patterns:

slow walk, fast walk, incline walk and walking with a ball. All subjects are captured using

six high resolution color cameras distributed evenly around the treadmill. Each sequence

is 11 seconds long, recorded at 30 frames per second. For concrete testing, we used the

silhouette sequences of one gait cycle for slow walk of three persons(subject 02, 06 and 13),

which are captured from a side view (camera vr03 7). The images are resized to 80 × 61.

Some examples of the silhouette images are shown in Figure 4.2.

(a) COIL-100

(b) COIL-100

Figure 4-1.: Databases examples

Three types of experiments are performed in order to validate the proposed methodology

and show its advantages against other frameworks. Firstly, we use the selected objects of

COIL-100 with a same amount of observations per set (n1 = n2 = n3 = 72), and equal input

dimensions (p1 = p2 = p3 = 16384). In this case, the number of nearest neighbors is fixed as

k = 3. In Figure 4-2 are shown the results for this experiment.

For the second experiment, we use the CMU Mobo database, which leads input samples

per manifold of different sizes: n1 = 36, n2 = 40, n3 = 38 and p1 = p2 = p3 = 4880 according

to each subject. The number of neighbors is set to k = 2. In order to test the algorithms on

a dataset that contains multiple manifolds with high-variability in the input sample sizes,

we use the COIL-100 but performing an uniform sampling of the observations. Therefore,

input spaces with n1 = 72, n2 = 36, n3 = 18 and p1 = p2 = p3 = 16384 were obtained. Here,

the number of nearest neighbors are fixed as k1 = 4, k2 = 2, k3 = 1, taking in to account


that is not suitable to set the same value of k because the sample size is so different for each

manifold. The results for the second experiment are presented in Figure 4-3 and 4-4.

Finally, the third experiment aims to validate the proposed methodology for analyzing

datasets with different amount of observations and different input dimensions (or image

resolution). For this purpose, we employ the COIL-100 performing an uniform sampling

of the observations, and resizing the images. Thence, the obtained input spaces have the

following characteristics: n1 = 72, n2 = 36, n3 = 18 and p1 = 16384, p2 = 8100, p3 = 2784.

This experiment is only carried out for the proposed approach, due to the other techniques

are not able to work with different input dimensions (Figure 4-5).

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

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(c) Proposed approach - MML (d) Matrix A

Figure 4-2.: Three COIL-100 objects, equal amount of observations

4.3 Discussion 43

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Figure 4-3.: Three COIL-100 objects, different amount of observations

4.3. Discussion

According to the results presented in Figures 4.2(a), 4.3(a) and 4.4(a), the traditional LEM is

not able to find the correspondence among different datasets which are related to a common

underlying data structure. For all the provided experiments, LEM performs a clustering

of points for each manifold and it does not preserve the local or global properties of the

high-dimensional data. This algorithm can not find a low-dimensional representation that

unfolds the underlying data structure from multiple manifolds, due to the weight matrix W

in LEM is computed only considering pixel intensity similarities among frames, which in this

case produce the above mentioned clusters. The performance is the same for the different

experimental conditions and both databases.

Again, taking into the account the attained results with the LJM technique (Figures 4.2(b),


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Figure 4-4.: Three Gait subjects, different amount of observations

4.3(b), and 4.4(b)), it can be seen how it attempts to find a correspondence among datasets,

nevertheless the intrinsic geometry data structure of the phenomenon (object motion) is lost.

More precisely, for the COIL-100 database, the dynamic of the rotation is not reflected in

the embedded space. Similar results are obtained for gait analysis in the Mobo database,

although LJM tries to reveal the elliptical motion shape, it is not able to conserve a soft

correspondence among samples. Note that the application of LJM technique is limited to

analyze frames of video sharing a similar geometry, due to Ucb is inferred in the high-

dimensional space (pixels frame comparison). Overall, the LJM method can not properly

learn the relationships among objects/subjects performing the same activity, it just develops

well when the analyzed manifolds are similar in appearance, which limits the applicability

of the technique.

4.3 Discussion 45

(a) Multi Manifold Embedding

(b) Matrix A

Figure 4-5.: Different amount of observations and input dimensions (image resolution)

3.5 Comparison Results of Nonlinear Dimensionality Reduction ...

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