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Linear and Fisher Separability of Random Points in the D Linear and Fisher Separability of Random Points in the d-dimensional Spherical Layer S.V.Sidorov N.Yu.Zolotykh Institute of Information Technologies, Mathematics and Mechanics Lobachevsky State University of Nizhni Novgorod Nizhni Novgorod, Russia Email: [email protected], [email protected] Abstract—Stochastic separation theorems play important role [8]. Wider classes of distributions (including non-i.i.d.) are in high-dimensional data analysis and machine learning. It turns considered in [3]. Roughly speaking, these classes consist of out that in high dimension any point of a random set of distributions without sharp peaks in sets with exponentially points can be separated from other points by a hyperplane with high probability even if the number of points is exponential small volume. Estimates for product distributions in the cube in terms of dimension. This and similar facts can be used and the standard normal distribution is obtained in [9]. for constructing correctors for artificial intelligent systems, for We note that there are many algorithms for constructing a determining an intrinsic dimension of data and for explaining functional separating a point from all other points in a data various natural intelligence phenomena. In this paper, we refine set (Fisher linear discriminant, linear programming algorithm, the estimations for the number of points and for the probability in stochastic separation theorems, thereby strengthening some support vector machine, Rosenblatt perceptron etc.). Among results obtained earlier. We propose the boundaries for linear all these methods the computationally cheapest is Fisher and Fisher separability, when the points are drawn randomly, discriminant [2]. Other advantages of the Fisher discriminant independently and uniformly from a d-dimensional spherical are its simplicity and the robustness. layer. These results allow us to better outline the applicability limits of the stochastic separation theorems in applications. The papers [1]–[3], [7] deal with only Fisher separability, Index Terms—stochastic separation theorems, random points, whereas [8] considered a (more general) linear separability. 1-convex set, linear separability, Fisher separability, Fisher linear A comparison of the estimations for linear and Fisher separa- discriminant bility allows us to clarify the applicability boundary of these methods, namely, to answer the question, for what d and n it I. INTRODUCTION suffices to use only Fisher separability and there is no need to Recently, stochastic separation theorems [1] have been search a more sophisticated linear discriminant. widely used in machine learning for constructing correctors In [8] there were obtained estimations for the cardinality and ensembles of correctors of artificial intelligence systems of the set of points that guarantee its linear separability when [2], [3], for determining the intrinsic dimension of data sets the points are drawn randomly, independently and uniformly [4] and for explaining various natural intelligence phenomena, from a d-dimensional spherical layer and from the unit cube. such as grandmother’s neuron [5] etc. These results give more accurate estimates than the bounds If the dimension of the data is high, then any sample of obtained in [1], [7] for Fisher separability. Here we give the data set can be separated from all other samples by a even more precise estimations for the number of points in hyperplane (or even Fisher discriminant – as a special case) the spherical layer to guarantee their linear separability. Also, with a probability close to 1 even the number of samples we report the results of computational experiments comparing arXiv:2002.01306v2 [math.PR] 18 Apr 2020 is exponential in terms of dimension. So, high-dimensional the theoretical estimations for the probability of the linear datasets exhibit fairly simple geometric properties. Due to the and Fisher separabilities with the corresponding experimental applications mentioned above the theorems of such kind can frequencies and discuss them. be considered as a manifestation of so called the blessing of dimensionality phenomenon [1], [6]. II. DEFINITIONS In its usual form a stochastic separation theorem is formu- A point X Rd is linearly separable from the set M Rd lated as follows. A random n-element set in Rd is linearly ∈ ⊂ bd if there exists a hyperplane separated X from M, i.e. there separable with probability p> 1 ϑ, if n<ae . The exact exists A Rd such that (A ,X) > (A , Y ) for all Y M. form of the exponential function− depends on the probability X ∈ X X ∈ A point X Rd is Fisher separable from the set M Rd distribution that determines how the random set is drawn, if the inequality∈ (X, Y ) < (X,X) holds for all Y M⊂[2], and on the constant ( ). In particular, uniform ϑ 0 <ϑ< 1 [3]. ∈ distributions with different support are considered in [1], [7], A set of points X ,...,X Rd is called 1-convex [10] { 1 n}⊂ The work is supported by the Ministry of Education and Science of Russian or linearly separable [1] if any point Xi is linearly separable Federation (project 14.Y26.31.0022). from all other points in the set, or, in other words, the set of vertices of their convex hull, conv(X1,...,Xn), coincides The authors of [1], [7] formulate their results for linearly with X ,...,X . separable sets of points, but in fact in the proofs they used { 1 n} The set X ,...,X is called Fisher separable if that the sets are only Fisher separable. { 1 n} (Xi,Xj) < (Xi,Xi) for all i, j, such that i = j [2], [3]. Note that all estimates (1)–(5) require 0 <r< 1 with strong Fisher separability implies linear separability6 but not vice inequality. This means that they are inapplicable for (maybe versa (even if the set is centered and normalized to unit the most interesting) case r =0. variance). Thus, if M Rd is a random set of points from The both estimates (1), (5) are exponentially dependent on a certain probability distribution,⊂ then the probability that M d for fixed r, ϑ and the estimate (1) is weaker than (5) (see is linearly separable is not less than the probability that M is Section V). Fisher separable. In [8] it was proved that if 0 r< 1, 0 <ϑ< 1, d ≤ Let Bd = X R : X 1 be the d-dimensional { ∈ k k ≤ } n< ϑ2d(1 rd), unit ball centered at the origin ( X means Euclidean norm), − is the -dimensional ball ofk radiusk centered at the rBd d r< 1 then P (d, r, n) > 1 ϑ.qHere we improve this bound (see origin. − Corollary 2) and also give the estimates for P1(d, r, n) and Let M = X ,...,X be the set of points chosen n { 1 n} P (d, r, n) and compare them with known estimates (2), (4) randomly, independently, according to the uniform distribution F F for P1 (d, r, n) and P (d, r, n). on the spherical layer Bd rBd. Denote by P (d, r, n) the \ F probability that Mn is linearly separable, and by P (d, r, n) IV. NEW RESULTS the probability that Mn is Fisher separable. The following theorem gives a probability of the linear Denote by P1(d, r, n) the probability that a random point separability of a random point from a random n-element set chosen according to the uniform distribution on Bd rBd is Mn = X1,...,Xn in Bd rBd. The proof uses an approach F \ { } \ separable from Mn, and by P1 (d, r, n) the probability that a borrowed from [10], [11]. random point is Fisher separable from Mn. Theorem 1. Let 0 r< 1, d N. Then ≤ ∈ III. PREVIOUS WORKS n P (d, r, n) > 1 . (6) 1 − 2d In [1] it was shown (among other results) that for all r, ϑ, n, d, where 0 <r< 1, 0 <ϑ< 1, d N, if Proof. A random point Y is linearly separable from Mn = ∈ X1,...,Xn if and only if Y / conv(Mn). Denote this d 2 d/2 { } ∈ r 2ϑ(1 r ) event by C. Thus P1(d, r, n) = P(C). Let us find the upper n< 1+ − 1 , (1) √ 2 r2d − bound for the probability of the event C. This event means that 1 r r ! − the point Y belongs to the convex hull of Mn. Since the points then Mn is Fisher separable with a probability greater than in Mn have the uniform distribution, then the probability of 1 ϑ, i.e. P F (d, r, n) > 1 ϑ. C is −The following statements− are proved in [7]. Vol conv(Mn) conv(Mn) rBd For all r, where 0 <r< 1, and for any d N P (C)= \ ∩ . • ∈ n Vol(Bd) Vol(rBd) (1 r2)d/2 − P F (d, r, n) > (1 rd) 1 − . (2) Let us estimate the numerator of this fraction. We denote 1 − − 2 by Si the ball with center at the origin and with the diameter For all r, ϑ, where 0 <r< 1, 0 <ϑ< 1, and for OXi. We denote by Ti the ball with center at the origin and • sufficiently large d, if with the diameter r inside the ball Si. Then ϑ n< (3) n (1 r2)d/2 conv(M ) (conv(M ) rB ) (S T ) − n \ n ∩ d ⊆ i \ i F i=1 then P1 (d, r, n) > 1 ϑ. [ For all r, where 0 <r<− 1, and for any d N and • ∈ n n (1 r2)d/2 Vol conv(M ) conv(M ) rB Vol(S T )= P F (d, r, n) > (1 rd) 1 (n 1) − . n \ n ∩ d ≤ i \ i − − − 2 i=1 (4) X n n r d For all r, ϑ, where 0 <r < 1, 0 <ϑ< 1 and for = (Vol(S ) Vol(T )) = Vol(S ) γ • i − i i − d 2 ≤ sufficiently large d, if i=1 i=1 X X n √ϑ 1 d r d nγ (1 rd) n< (5) γ γ = d − , (1 r2)d/4 ≤ d 2 − d 2 2d i=1 ! − X then P F (d, r, n) > 1 ϑ.
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