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In document Acta 2502 y (Pldal 143-152)

We presented a deep neural network based speech de-identification method that can map vocoder features of human speech to those of a generic TTS engine with little or minimal loss of sound quality using the TIMIT data set. The novelty of our scheme is that de-identification is based on speech-text sample pairs, which are widely available in the speech processing community. In the resulting signal,

2https://people.inf.elte.hu/foauaai/deidentification

Table 2: Results of the perceptual listening experiments. We report the average and the standard deviation of the identification accuracy.

Task # of

subj.

# of samp.

Accuracy mean±std

Random choice A-not-B

22

20 0.56±0.15 0.5

Female/Male (2AFC) 15 0.51±0.15 0.5

# of Speakers 6 0 0.16

(6AFC) 6 0.18 0.16

the identity of the speaker is concealed, as confirmed by our perceptual listening experiments.

A limitation of our technique is that the dynamics of the original speaker are inherited due to the application of DTW. We hypothesize that this problem may be alleviated by applying DTW in the loss function of the deep network. We leave such studies to future work.

Our technique enables privacy-aware speech recognition for adults. The pro- posed method is lightweight and can be used for collecting de-identified databases when the privacy of the user is important, for example in cloud-based speech services or in medical records. The fact that our method requires only speech- transcript sample pairs is a very promising aspect for deep learning, which requires large and high quality databases.

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Toolset for Supporting the Research of Lattice Based Number Expansions

P´ eter Hudoba

ab

and Attila Kov´ acs

ac

Abstract

The world of generalized number systems contains many challenging areas.

Computer experiments often support the theoretical research. In this paper we introduce a toolset that helps to analyze some properties of lattice based number expansions. The toolset is able to (1) analyze the expansions, (2) decide the number system property, (3) classify and visualize the periodic points.

The toolset is implemented in Python, published alongside with a database that stores plenty of special expansions, and is able to store the custom prop- erties like signature, operator eigenvalues, etc. Researchers can connect to the server and request/upload data, or perform experiments on them.

We present an introductory usage of the toolset and detail some results that has been observed by the toolset. The toolset can be downloaded from http://numsys.infodomain.

1 Introduction

The generalization of positional number representations to a wide range of digit sets or to higher dimensions is a fascinating story. Gr¨unwald (1885) investigated negative-based, Kempner (1936), Knuth (1960), Khmelnik (1964), Penney (1965) complex-based systems. From the 70’s K´atai, B. Kov´acs, K¨ornyei, Peth˝o (the

“Hungarian school”) and Gilbert examined systematically the radix extensions in algebraic number fields. In the 90’s the topological aspects of radix representa- tions were studied by Bandt, Indlekofer, J´arai, K´atai, Lagarias, Wang, Vince, and later by Akiyama, Thuswaldner and others. The canonical number representation was generalized toarbitrary polynomial systems by Peth˝o (1989), and investigated later extensively by many authors (incl. Akiyama, Brunotte, Kov´acs, Peth˝o, Rao, Scheicher, Thuswaldner). The number system concept in general lattices was in- vestigated first by Vince (1993). Thealgorithmic aspectsof canonical (polynomial) systems was initiated by Brunotte (2001) and for general lattices by the second

aotv¨os Lor´and University, Budapest, Hungary

bE-mail:peter.hudoba@inf.elte.hu, ORCID:0000-0001-5810-4193

cE-mail:attila@inf.elte.hu, ORCID:0000-0002-1858-7618

DOI:10.14232/actacyb.289524

author (2000). Recently, a special type of radix systems (SRS) studied in length by Thuswaldner and his co-workers (the “Austrian school”).

2 Preliminaries

Let Λ be a lattice in Rn and let M : Λ Λ be a linear operator such that det(M)= 0. Let furthermore 0 ∈D⊆Λ be a finite subset. Lattices can be seen as finitely generated free Abelian groups and have many significant applications in pure mathematics (Lie algebras, number theory and group theory), in applied mathematics (coding theory, cryptography) because of conjectured computational hardness of several lattice problems, and are used in various ways in the physical sciences. We note that the number system research in general lattices comprises also the orders.

Definition 1. The triple, M, D) is called a number system (GNS) if every elementxof Λhas a unique, finite representation of the form

x= L i=0

Midi ,

wheredi∈D andL∈Z(L+ 1is the length of the expansion).

HereM is called thebase andD is thedigit set. It is easy to see that similarity preserves the number system property, i.e., ifM1andM2are similar via the matrix Q then (Λ, M1, D) is a number system if and only if (QΛ, M2, QD) is a number system at the same time. If we change the basis in Λ a similar integer matrix can be obtained, hence, there is no loss of generality in assuming that M is integral acting on the latticeZn. If two elements of Λ are in the same coset of the factor group Λ/MΛ then they are said to becongruentmoduloM. The following theorem gives a necessary condition for the number system property.

Theorem 1. If, M, D) is a number system, then (1) D must be a full residue system modulo M, (2) M must be expansive, and (3) det(In−M) =±1. (unit condition). If a system fulfils the first two conditions then it is called a radix system.

We note that the theorem in this form was stated first in [9] but it was well- known and used much earlier by K´atai and Vince. The full residue system property can be decided easily using Smith normal form [8]. Algorithms, that calculate the eigenvalues of M exactly in a finite number of steps exist only for a few special classes of matrices. For general matrices, iterative algorithms produce approximate solutions. In polynomial systems, where M is the companion of a monic inte- ger polynomialf, deciding the Schur or Hurwitz stability of f is computationally equivalent with the expansivity check. Verification of the unit condition is trivial.

Write ϕ : Λ Λ, x ϕ M1(x−d) for the unique d D satisfying x d (modM). Since M1 is contractive andD is finite, there exists a norm . on Λ and a constant C such that the orbit of every x Λ eventually enters the finite

set S ={x∈Λ| x< C} after repeated application of ϕ. This means that the sequencex, ϕ(x), ϕ2(x), . . . is eventually periodic for allx∈Λ. Clearly, (Λ, M, D) is a number system iff for everyx∈Λ the orbit ofxeventually reaches 0. A point pis calledperiodicifϕk(p) =pfor somek >0. The orbit of a periodic pointpis a cycle. The set of all periodic points is denoted byP. Thesignature [l1, l2, . . . , lω] of a radix system is a finite sequence of non-negative integers in which the periodic structure P consists of #li cycles with period lengthi (1 ≤i ≤ω). Clearly, the signature of a number system isSig= [1].

The following problem classes are in the mainstream of the research.

• For a given (Λ, M, D) thedecision problem asks if the triple forms a number system or not.

• For a given (Λ, M, D) theclassification problem means finding all cycles (wit- nesses).

• Theparametrization problemmeans finding parametrized families of number systems.

• Theconstruction problem aims at constructing a digit setD toM for which (Λ, M, D) is a number system. In general, construct a digit setDtoM such that (Λ, M, D) satisfies a given signature.

We note that the algorithmic complexity of the decision and classification problems are still unknown.

Thefundamental domain or set of “fractions” in (Λ, M, D) can be defined as H =

'

i=1

Midi:di∈D (

Rn.

Theorem 2. (a)H is compact. (b)x∈ P ⇔x∈ −H.

The compactness was proved by many authors (see e.g. Vince [15]). The part of (b) was proved in [9]. The other direction is obvious as well, otherwise 0 would have at least two different representations.

The theorem means that in order to determine the periodic points it is enough to localize the lattice points in−H. There are two different approaches to overcome this problem: the IFS-method (see [8, 10]), and the covering method (see [8, 4]), which was optimized by the authors [6]. The idea of the latter is that we can put the compact set−H into a boxB in which the integer elements are easily enumerable.

Then, we can compute the pairs (x, ϕ(x)) for allx∈B, and finally, we determine the cycles applying one of the cycle finding methods.

There are other algorithms for solving the decision/classification problems.

Based on the method of Vince [15], Brunotte [2] described a digit-propagation algorithm for polynomial systems with canonical digits. Later, his method was gen- eralized for arbitrary operators and digit sets [4]. The shortcoming of this method

is the sequential nature of the digit propagation, however, there is an algorithmic attempt to overcome this disadvantage [14].

Let f(x) = a0 +a1x+a2x2+· · ·+an1xn1 +xn be an integer (monic) polynomial. Let us denote the factor ring Z[x]/(f) by Λf. Then Λf is a lattice and all the problems regarding number expansions in Λf can be formulated inZn, where M is the companion of f. If f is irreducible then Λf is isomorphic toZ[θ]

where f(θ) = 0 in an appropriate extension of Q. Hence, if the digit set D is restricted to be a set of non-negative numbers D ={0,1, . . . |a0 | −1}, we get a straightforward generalization of the traditional number systems inZ. In this case the digit set is calledcanonical. If the radix system (Λf, θ, D) satisfies the unique representation property of Definition 1 with some canonical digit set D then it is called a canonical number system (CNS). The notion of canonical digit sets can be extended to form a j-canonical set Dj = {0, ej, . . . ,(| a0 | −1)ej} ⊂ Zn (ej

is thejth unit vector) [8]. There exists a canonical number system in OK – the ring of integers of the algebraic number fieldK – if and only if there is a power integral basis inOK [12]. We note that canonical digit sets may or may not exist in different lattices and canonicity depends on the chosen basis. Thesymmetric digit set is formed by those integer multiples ofej which are closest to the origin, and was introduced by K´atai [7]. The adjoint digit set consists of those lattice points which are in det(M))

21,,12

. Thedense digit set — in which each digit has the smallest norm in its residue class — were introduced and used extensively by the second author. We note that the adjoint set is a dense one in a special basis.

In document Acta 2502 y (Pldal 143-152)