CC BY
12
2018, Т. 160, кн. 2 С. 220-228
УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА. СЕРИЯ ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ
ISSN 2541-7746 (Print) ISSN 2500-2198 (Online)
UDK 519.23
ESTIMATION OF SMOOTH VECTOR FIELDS ON MANIFOLDS BY OPTIMIZATION ON STIEFEL GROUP
E.N. Abramova, Yu.A. Yanovichb'c'a
aNational Research University Higher School of Economics, Moscow, 101000 Russia bSkolkovo Institute of Science and Technology, Moscow, 143026 Russia cKharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, 127051 Russia
Abstract
Real data are usually characterized by high dimensionality. However, real data obtained from real sources, due to the presence of various dependencies between data points and limitations on their possible values, form, as a rule, form a small part of the high-dimensional space of observations. The most common model is based on the hypothesis that data lie on or near a manifold of a smaller dimension. This assumption is called the manifold hypothesis, and inference and calculations under it are called manifold learning.
Grassmann & Stiefel eigenmaps is a manifold learning algorithm. One of its subproblems has been considered in the paper: estimation of smooth vector fields by optimization on the Stiefel group. A two-step algorithm has been introduced to solve the problem. Numerical experiments with artificial data have been performed.
Keywords: manifold learning, dimensionality reduction, vector field estimation, optimization on Stiefel manifold
Introduction
The concept of Big Data implies not only a large sample size, but also high dimensionality. Thus, real data usually have a very high dimensionality, for example, the dimensionality of digital black and white photos is equal to the number of pixels (up to 10 000 pixels), while brain images obtained per second with functional magnetic resonance imaging have a size about 1.5 million. However, many traditional methods and algorithms become inefficient or simply inoperable for high-dimensional data, and this phenomenon is called the curse of dimensionality [1]. D. Donoho, a famous statistician, said in 2000 at the Conference on Mathematical Challenges of the 21st Century: "We can say with complete confidence that in the coming century high-dimensional data analysis will be a very significant activity, and completely new methods of high-dimensional data analysis will be developed; we just do not know what they are yet" [2].
However, real data obtained from real sources, due to the presence of various dependencies between data points and limitations on their possible values, take, as a rule, a small part of the high-dimensional space of observations. For example, the set of all black and white portraits of human faces with the original dimension of the order of hundreds of thousands has an intrinsic dimension of not more than 100. The consequence of the low intrinsic dimension is the possibility of constructing a low-dimensional parametrization of such data with the minimal loss of information contained in it. Therefore, many algorithms for high-dimensional data processing begin with the problem of dimensionality reduction, which results in low-dimensional descriptions of such data.
The most common model for describing such data is the hypothesis that data lie on or near a manifold M of a smaller dimension. This assumption is called the manifold hypothesis [3], and inference and calculations under it are called manifold learning [4,5].
At the moment, many manifold learning algorithms have been developed: IsoMap [6], locally linear embedding [7], local tangent space alignment [8], Laplacian eigenmaps [9, 10], Hessian eigenmaps [11], Grassmann& Stiefel eigenmaps (GSE) [12, 13], etc. It is the GSE calculation aspects that are considered in the paper.
1. Manifold learning. Grassman—Stiefel eigenmaps
1.1. Manifold learning. Manifold embedding. Manifold learning problem is to find the mapping h : M C Rp ^ Y C Rq given X = (X1, .. .,XN) £ M C Rp, such that Y shows the structure of high-dimensional data, where M is a manifold with dim M = q. The points Xi,...,XN are assumed to be independent identically distributed (iid) random variables from a continuous measure ^ with supp^ = M. In addition, there should exist (and it should be estimated) the reverse map g: Y ^ Rp and the approximation error — g(h(x))\\ should be small.
Assuming that the manifold M can be covered by single coordinate chart B, we can define coordinates (bi,..., bq) that represent our manifold. Thus, we can write in Rp : X = f (b), X' = f (b') and for two close points X, X' consider the linear approximation:
X' = Jf (b)(b' — b) + X, where Jf (b) is the Jacobian matrix. Thus, we reformulate our problem as follows:
E||(X' — X) — Jf(b)(b' — b)f ^ mi^ (1)
for all close points (this notion will be defined further). GSE separates this problem into two parts: the first task is to estimate the matrix Jf = Jf (f-1(X)), the second one is to estimate coordinates b for a given Jf . The latter problem is ordinary regression, while the former one is of interest.
1.2. Local description. Neighborhoods construction. The Jacobian matrix Jf(b) = (dfi/dbj), i = l,...,p, j = l,...,q, defines a tangent space TX(M) at point X = f (b), where TX(M) is a q-dimensional linear subspace. The tangent spaces can be considered as elements of the Grassmann manifold Grass(p,q) consisting of all q-dimensional linear subspaces in Rp.
Every neighborhood of each point can be approximated by the tangent space (1) (linear subspace of Rp) as M is smooth. So, we should define the "linear" neighborhood. For each sample point Xn £ X, local neighborhood U(Xn) = {X1,... ,Xk(n)} consisting of k-near sample points is constructed. Typical examples of U(Xn) are U(Xn,e), the neighborhood of Xn consisting of sample points that from e-ball centered at Xn, and U(Xn,k) consisting of k-nearest to the Xn sample points; in our calculations, the latter one is used. These neighborhoods determine "Euclidean" kernels K (Xi,Xj) = I (Xj £ U (Xi)), where I (A) is the indicator of event A, i.e., I (A) = 1 if A occurred, and I (A) = 0 otherwise.
Then, defining these linear subspaces is the standard manifold learning problem, which is solved by principal component analysis (PCA). Thus, at each point we search the directions of maximum variation, defining orthogonal vectors. These vectors are joined in matrix Qpca £ Rpxq, which is considered to be a projector from the q-dimensional tangent space into Rp. It is worth noting that Qpca are elements of the Stiefel manifold Stief (p,q) - the set of all q-frames in Rp: QPCaQPCa = Iq.
1.3. Global description. We construct q-dimensional subspaces at each point Xi as an approximation of the tangent bundle of a differentiable manifold. However, our goal is to set a coordinate system on a manifold. Obviously, the PCA algorithm does not provide coordinated frames. PCA estimation consistency and its rates are obtained, for example, in [14, 15]. Thus, these frames should be rearranged in some tanged fields H(Xi) and Span(QPCA(X.j)) = Span(H(Xi)), where Span(A) is a linear hull of columns A1,..., Am of the matrix A = (Ai,..., Am). Let v G Rqxq be some non-degenerate transformation in the tangent. Then, the frame Qpca is transformed into the vector field Qpca • v from the same tangent space. Generally, for each point we write:
Hi = QpcA(Xi) • vi (2)
wherevi = v(Xi) is a non-degenerate transformation of q-basis into q-dimensional subspace, Hi = H(Xi) is a vector field at point Xi.
While defining the distance between two vectors by the Euclidean norm ||.||2, it is natural to set the distance between two vector fields by the Frobenius matrix norm
IH - Hj I
^Tr ((Hi - Hj)T(Hi - Hj)). (3)
Then, to measure the closeness between points (and, because of differentiability, also between tangent spaces), we define the function K(Xi,Xj) = Kij. Finally, we face the next optimization problem:
A(H„) = £ Kij IIHi - Hj IF ^ min (4)
ij||F ^ mm
i=1
subject toVi = 1,... ,n rank(vi) = q, SpanHi = SpanQ pca (Xi), HT Hi = Iq.
We have {Hi}rn=1 C Stief (p,q) as SpanHi = SpanQPCA(Xi) and Hf Hi = Iq for i =l,...,n. The set of {Hi}n=1 could be explicitly extended H (X) for all X gM [12]. So, the solution of (4) with its extension is a smooth vector fields estimation on manifold by optimization on the Stiefel manifold (group). The problem (4) was introduced in [16], but there was only zero-order heuristic solution. Furthermore, our problem was solved in [17] by the direct iterative procedure. In this paper, we introduce a theoretically motivated one.
The sample X should be "dense" on M to get a good global vector field estimation. Such a property is guarantied with high probability for good enough manifolds [15, 18].
2. Algorithm
Orthogonal matrix vi G O(q) rotates the q-frame Qpca in Rp, such that it finally defines the same subspace. Thus, we get the following variation of functional (4):
n
A = V" KijIIQiVi - QjvjIF ^ min (5)
{Vi},i=l...n
i= 1
s.t. VTVi = Iq,
where Qi = QpcA(Xi). This is an optimization problem with orthogonal constraints: optimization on Lie group O(q). Nevertheless, the implementation of such optimization
is impossible, because O(q) has two connected components (simply put matrices with determinant +1 and — 1). Thus, optimization is feasible on special orthogonal group, i.e., proper rotation of q-frame. Yet, such optimization with our PCA-frames is degenerate, because for two infinitely close points Xi, Xj the distance \Hi —Hj||_p in (4) will be non-zero. To solve this problem, we must obtain an algorithm defining self-orientation of bases on manifold and, then, defining the orientation on manifold. Let us remind here that the manifold is orientable if there exists the set of maps (Ua,4>a), such that the determinant of the Jacobian matrix is positive for each a, 3; Ua p| Up = 0.
Proposition 1. Let the manifold M be orientable. Qi,Qj £ Rpxq are the matrices defining q -orthogonal unit vectors in the p -dimensional space for some close points Xi, Xj Then, the condition det(QTQj) > 0 for every close points Xi, Xj sets an oriented atlas of manifold M .
Condition det(QTQj) > 0 is easy to check for constructed frames (matrices QjQj are Rqxq, q no more than 10). Via this fact, we can make an inductive algorithm defining the orientation on the manifold. Let Ak = {Xi1 ,...,Xik} be the set of k points, where the right orientation is defined. In the first step, we take random point Xi1 and set the right orientation at this point Qi1; A1 = {Xi1}. In the k-th step, we get the set of points Ak = {Xi1,... ,Xik }. Then, we find the closest point among X\Ak to this set (Xik+1 = arg min HZ — YH2); Y £ Ak . Following that, we check the condition
Qjk+1 Qj > 0, where Xj £ Ak is the closest point to Xik+1 . If det(Qfk+1 Qj) < 0, then two columns in Qik+1 should be reversed, otherwise Qik+1 gives the right orientation, in both cases Ak+1 = Aky]{Xik+1}. Finally, we get the whole set of points (X1,... ,XN) and the set of oriented frames (Q1,... ,QN).
By transforming functional (5), consider QTQi = Iq, we get the following optimization problem:
n
A = — £ Kij ■ Tr (vT(QjQj)vj) ^ mi? (6)
A brief scheme of the algorithm can be presented as follows:
A. constructing graph Kij ; Kij = 1 if Xj is among k -closest points of Xi , otherwise Kij = 0;
B. constructing frames Qpca by PCA at each point of sample;
C. initialization of frames Qpca - defining orientation of manifold;
D. solving optimization problem (6) by Trust Region Method [19].
3. Numerical experiments
The half of two- and four-dimensional spheres S^1>o C R3, §^>0 C R5 and cylinder S1 x [0,1] are considered as manifold M. All samples are iid uniformly distributed.
One can see that, as a result of optimization, initialization converges to the ideal value of —2 fast and vector fields are visually smooth. The A values without optimization are higher and the vector fields are not smooth.
Sample generation and steps 1-3 of the algorithm are implemented in R with the standard packages. Step 4 of optimization is performed in Matlab using the Manopt package for optimization on manifolds [20]. The derivatives and the Hessian matrix were theoretically calculated for the functional (6) and were used as a part of input
Fig. 1. A as a function of log10(n/1QQ) for the 2D sphere
Fig. 2. A as a function of log10(n/1QQ) for the 4D sphere
Cylinder, optimized functional
log(n/100)
Fig. 3. A as a function of log10(n/1QQ) for the cylinder
for manopt. The functional (6) was scaled to present results for different sample sizes, dimensions and neighbor numbers k: A = 2(Nqk)-1A. For two close points Xi, Xj, the function is q-1Tr (HTHj) ^ 1. Therefore, functional A should be close to —2. The value of functional A as a function of log10(N/100) is calculated for sample sizes N = 100 • 2m, m = 0,..., 4 for the cylinder and two-dimensional sphere and m = 0,..., 3 for the four-dimensional sphere. For each sample size, 10 samples were calculated. In Figs. 1, 2 and 3, the mean value of functional (6) and its mean +/— standard deviation are shown.
In Figs. 4 and 5, the optimized vector fields (a and c) and vector fields without optimization (b and d) for the 2D sphere and cylinder are indicated.
Conclusions
The optimization aspects of vector field alignment on an unknown manifold were considered. It is an important part of the Grassmann & Stiefel eigenmaps algorithm for the manifold learning problem.
The algorithm for vector field alignment was introduced. The algorithm consists of two steps: bases initialization at the same oriented atlas and Stiefel group optimization. The numerical experiments with artificial data were performed. The proposed algorithm shows good results.
Acknowledgements. The study by Yu.A. Yanovich was supported by the Russian Science Foundation (project no. 14-50-00150).
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Received
December 8, 2017
Abramov Evgeny Nikolayevich, Graduate Student of the Faculty of Computer Science National Research University "Higher School of Economics"
ul. Myasnitskaya, 20, Moscow, 101000 Russia E-mail: [email protected]
Yanovich Yury Alexandrovich, Candidate of Physical and Mathematical Sciences, Researcher of the Center for Computational and Data-Intensive Science and Engineering; Researcher of the Intelligent Data Analysis and Predictive Modeling Laboratory; Lecturer of the Faculty of Computer Science
Skolkovo Institute of Science and Technology
ul. Nobelya, 3, Territory of the Innovation Center "Skolkovo", Moscow, 143026 Russia Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences
Bolshoy Karetny pereulok, 19, str. 1, Moscow, 127051 Russia National Research University "Higher School of Economics"
ul. Myasnitskaya, 20, Moscow, 101000 Russia E-mail: [email protected]
УДК 519.23
Оценивание гладких векторных полей на многообразии с помощью оптимизации на специальной ортогональной группе
Е.Н. Абрамов1, Ю.А. Янович2'3г1
1 Национальный исследовательский университет «Высшая школа экономики», г. Москва, 101000, Россия 2 Сколковский институт науки и технологий, г. Москва, 143026, Россия
3Институт проблем передачи информации Харкевича РАН, г. Москва, 127051, Россия
Аннотация
Зачастую данные, полученные из реальных источников, имеют высокую размерность. Однако часто, в силу возможного наличия зависимости между параметрами, данные занимают лишь малую часть высокоразмерного пространства. Самая общая модель описания таких закономерностей - предположение о том, что данные лежат на или около многообразия меньшей размерности. Такое предположение называется гипотезой многообразия, область применения такой гипотезы - обучение на многообразии.
Вложения Грассмана-Штифеля - один из алгоритмов обучения на многообразии, вариация которого представлена в работе: оценивание гладких векторных полей на многообразии с помощью оптимизации на специальной ортогональной группе. Представлен алгоритм для решения задачи, проведены численные эксперименты на искусственных данных.
Ключевые слова: обучение на многообразии, снижение размерности, оценивание векторных полей, оптимизация на ортогональной группе
Поступила в редакцию 08.12.17
Абрамов Евгений Николаевич, студент факультета компьютерных наук Национальный исследовательский университет «Высшая школа экономики»
ул. Мясницкая, д. 20, г. Москва, 101000, Россия E-mail: [email protected]
Янович Юрий Александрович, кандидат физико-математических наук, научный сотрудник Центра по научным и инженерным вычислительным технологиям для задач с большими массивами данных; научный сотрудник лаборатории интеллектуального анализа данных и предсказательного моделирования; старший преподаватель факультета компьютерных наук
Сколковский институт науки и технологий
ул. Нобеля, д. 3, Территория Инновационного Центра «Сколково», г. Москва, 143026, Россия
Институт проблем передачи информации им. А.А. Харкевича РАН
Большой Каретный переулок, д. 19, стр. 1, г. Москва, 127051, Россия Национальный исследовательский университет «Высшая школа экономики»
ул. Мясницкая, д. 20, г. Москва, 101000, Россия E-mail: [email protected]
I For citation: Abramov E.N., Yanovich Yu.A. Estimation of smooth vector fields ( on manifolds by optimization on Stiefel group. Uchenye Zapiski Kazanskogo Universiteta. \ Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 2, pp. 220-228.
/ Для цитирования: Abramov E.N., Yanovich Yu.A. Estimation of smooth vector fields ( on manifolds by optimization on Stiefel group // Учен. зап. Казан. ун-та. Сер. Физ.-ма-\ тем. науки. - 2018. - Т. 160, кн. 2. - С. 220-228.
