On a non-classical recognition problem

189

REFERENCES flows, Ann. Rev. 1. ORSZAG S.A. and ISRAELI M., Numerical simulation of viscous incompressible Fluid Mech., 6, 281-318, 1974. 2. ROACHE P.J., Computational Fluid Dynamics, Hermosa Publ., Alberquerque, N.M., 1976. 3. TOLSTYKH A.I., On implicit higher-order of accuracy schemes for systems of equations, Zh. Vychisl. Mat. mat. Fir., 21, 2, 339-354, 1981. 4. BONTOUX P., FORESTIER B. and ROUX B., Analysis of higher-order methods for the numerical simulation of confined flows, In: Proc. 6th Internat. Conf. Numer. Meth. Fluid Dynamics. Lecture Notes Phys. No.90, Springer, Berlin, 94-102, 1979. 5. ROUX B. et al., Optimization of Hermitian methods for Navier-Stokes equations in the vorticity and stream function formulation, In: Lecture Notes Math. No.771, Springer, Berlin, 450-468, 1979. 6. RUBIN S.G. and KHOSLA P.K., Higher-order numerical solutions using cubic splines, AIAA Journal, 14, 7, 851-858, 1976. 7. RUBIN S.G. and KHOSLA P.K., Polynomial interpolation methods for visous flow calculations, J. Comput. Phys. 24, 1, 217-244, 1977. 8. RUBIN S.G. and GRAVES R.A, Viscous flow solutions with cubic spline approximation, Comput. Fluids, 3, 1, l-36, 1975. 9. ZAV'YALOV YU.S., KVASOV B-1. and MIROSHNICHENKO V.L., Spline Function Methods (Metody splain-funktsii), Nauka, Moscow, 1980. 10. SAMARSKII A.A. and NIKOLAEV E.S., Methods of solving Finite Difference Equations (Metody reshenya setochnykh uravneniil, Nauka, Moscow, 1978. Translated

U.S.S.R.,

Printed

Comput.Maths.~ath.Phys.,Vo1.24,No.3,pp.189-193,1984 in Great Britain

by E.J.S.

0041-5553/84 $lO.OO+O.OO 0 1985 Pergamon Press Ltd.

ON A NON-CLASSICAL RECOGNITION PROBLEM* A.I. ZENKIN,

V.K. LEONT'EV

One of the models of recognition problems when the object represented in the form of pieced data is considered.

presented

is

Let A be a training table which is broken down in the usual manner into K classes, L,, ...(L I. In the standard recognition model introduced in /l/ a vector is presented for recognition, and a decision has to be taken concerning the affiliation of a to one of the classes L,. In the recognition model presented below, it is not the vector a itself which is presented (it is also incompletely defined) but only certain information concerning it which is included in a set of fragments constituting pieced information concerning the vector a. As the outcome it is necessary to take a decision concerning the affiliation of the vector represented by the set of its fragments to one of the classes L.. It should be noted that one of the mathematical problems associated with the model under consideration in this paper has been formulated in /2/. Let a-(a,, . . ..a.)=E”. Definition vector a

1.

Any subsequence

of the sequence

(a,,...,a,) is called a fragment

of the

Example 1. Let a=(l0110)WP. Then, in particular, p,-(lc@), ~~=(lO),~~-(lllO) will be fragments of it. These fragments coorespond to the following subsets of coordinates of the vecotr a: (1, 2, 51, (4. 5). (1, 3, 4, 51. We shall write the fact that p is a fragment of Q as: @=a. It is clear that each a=E” has 2" fragments any of which can be prescribed by a certain subset of a segment of the natural series N,={I,2,. . . , n}. Another method of defining a fragment is to assign a characteristic vector v to it, the units of which correspond to the numbers of the coordinates of the fragment. Example 2. If a-((10110), v-(01110), then the fragment to the vector Y appears thus: (a,~)-(011).

of the vector

a which

corresponds

(1)

It is readily seen that the non-commutative fragmentation operator (,) introduced by rule where (1) is distributive with respect to the first component, i.e. -, the addition is intended to be mod2. We shall Let V-{v,, . . . , v,) be a set defining the fragments, i.e.,
l21l.vychisl.~!at.mat.Fiz.,24,6.925-931.1984

190 Definition 2. Two finite sets of vectors (z,)and (y,} are said to be equal if the number of entries of any vector in one set IS equal to the number of entries of the same vector in the other set. Definition

3.

The vectors

a, PEE"

are called

{(a, vJ) =((a,

V-equivalent

if (2)

v,)),

where

(2) is to be understood in the sense of the equality of the sets. It is clear that the V-equivalence relationship introduced using (2) is transitive and therefore the whole cube E" is decomposed into non-intersecting classes of equivalent vectors Ri : E” = ;I R,, I=,

R,nR,=0,

i#j.

The fundamental problem. It is known that it A set of fragments (e,) is prescribed. was obtained by the application of a fragmentation operator with vectors from V=( V,....,V") to a certain vector a=E" . It is necessary to describe the equivalence class of a and to find methods for locating any representative of this class which are acceptable in so far as their difficulty is concerned. In prescribing the set of fragments (e,) it is not known whether a given fragRemark. ment e, is obtained under the action of just some vector vjeE . In the opposite case the problem becomes trivial. Let v,~lJ, e,=(e), We shall denote the set of solutions of the equation (x,v)=e,, subject to the condition that x=E” , by L(v,, c,). dimensional interval in E", the fixed elements Lemma 1. The set L(v,,e,) is an (n-/iv,lj) of which coincide with the units of v, and are completed by the coordinates of e,. Example 3. Let v,=(lllOO), e,=(lOO) I then L(v,,e,)=(lOOa@), i.e. L(v,, ej) dimensional interval of E”. Later we shall use the notation L (v,, ej)=B,,. Let Definition

4.

is a two Av=I]BuII.

The set

is called the group theoretic permanent (g.t.p) of the matrix A,. The summation here is carried out over all permuations {!,&...,N). (i,, , iN) of the set Hence the g.t.p is simply the permanent of a matrix in which the numerical operations of addition and multiplication are replaced by the corresponding group theoretic operations Of union and intersection. We shall call the value of the permanent of the matrix A "the cardinal number of the set per A” and denote it by lperA,[. Theorem 1. The V-equivalence class R. of a vector matrix A,, constructed using the sets (vi} and (e,).

a=E”

coincides

with the g.t.p. of

By definition the set (e8} of f ragments a of a vectors a is obtained Proof. application of vectors from V to this vector. Then, for a certain Let BEper A”.

the inclusion fiEBll,fl...flSN,iN is true, from which it follows i.e. {
that

by the permutation

(~,v,,)=et,, r=l,Z,...,N,

(3)

perAvsR,

follows from this. On the other hand, if the vectors a and p from E" are V-equivalent, then a permutation i.e. S exists such that (8,a,)=e,,,,,i=l,2, . . ..N. from which it follows that g~8,.~,,fl...ilE?,,,,,, R,cper

Confirmation

of Theorem

1 follows from

Av.

(2) and

(4) (4).

is called V-reducible Definition 5. A vector a=E” vector has a cardinal number of unity. Corollary 1. A vector a is V-reducible is satisfied where .A, is a matrix constructed

if the V-equivalence

class of this

when and only when the condition (perAvj=l using the vectors ((a, vi)). {vi) and

is v-reducible. Definition 6. A V-set is said to be complete if each vector a=E” Hence, each class of equivalence of any complete set contains just a single vector. Example

4.

It is clear that the set V={v=(i,..., 1)) is complete

since

(a,v)=a.

Example 5. Let n=2k+l. Then the vectors v,=(ll...lO...O) and v,=(OO...O1...1) form a complete set. In fact, when a=(a,,...,a,,) , we have that (a, v,>=(a,,...,ak), (a, vz)=(CL*+t,...,%). Since the reduction algorithm involves putting the fragment of smaller length in front ken-k,

191

and the remaining fragment is added to it from the right. Example 6 (see /2/l. The layer E,* is a complete set when khd2 Several methods for reducing a vector using a given set of fragments are reported below. Let A,==(Ju,,l(be a matrix the rows of which are the elements of a V-set and .4.=11rUll be a table the rows of which are the vectors {e) where (x,vO==ei,i=l,2, . . ..N and x=(r,, . . . ..t.). shall denote the number of entries of the coordinate z, in the i-th column of the We the two following propertable A, by u,~ and denote the set of rows of matrix A whichpossess ties by Ttj : 11 the j-th coordinate of any row from T,* is equal to unity; 2) each row from T, contains just i units among the first j coordinates. Lemma

2.

The equation

where \Tij)is the number of rows in class t~;,,r.)T~~\,

true.

T.,, is

we shall find a number of situations in which the coordinate zi apears in the Proof. i-th column of table A.. In order that the z, coordinate should be displaced into the i-th column, it is necessary that, in the case of a vector v which acts on X, the j-th coordinate coordinates of this vector just j-1 ones should be equal to unity and among the first i-1 is prescribed by the should be encountered. In fact, in this case the fragment e=(a,v> conditions u;=i, u,+. ..+v,_,=i-i, i.e. e=(zs,...ss,.,z,)=(Yt . . . y,_,y,f, where y,=z, which had to be proved. The following theorem immediatelyfollows from Lemma 2. Theorem

2.

Let t, be the sum of the elements

2

Z,rITUI--tc,

of the i-th column of matrix

A..

Then

i=i,2,...,r,

(5)

j-1

where

r is the number of columns

A..

in table

coroi1ary 2. If system (5) has just one solution right-hand sides ft,,..., 1,) then the set V is complete. We shall now apply this result to the case V-E,,‘, of weight k occur in the V-set. Theorem

3.

In the case when

V-E.*

the system

when

~,~er{O,$) for arbitrary

natural

i.e. to the case when all binary

sets

of equations

“--*+i

x

‘--L

L-S

z&J,-,cm-, =trr

i-4,2

,...,

(6)

k.

1-t

of the i-th column of

Here ti is the sum of the elements is satisfied. dimension C,LXk. First we shall prove the relationship

the matrix

d, of

In fact, in this case the matrix A. contains all binary sets containing just k unit coordinates as rows. Hence the choice of submatrices satisfying conditions 1) and 21 can be made in cjzi ways from the left and c"=$ ways from the right, which also proves the validity of (71. System (6) now immediatelyfollows from (7) and from the definition of the numbers k=n-1. As an illustration of the use of Theorem 3 we shall consider the case when such a value of k, expression (7) takes the form 2,(n-1)Sz,+,i--tr, i4,2,.

t,. With

. . , n-i.

(8)

System (8) can be used in the following manner to establish the vector x=(a,,...,a.) with n-l. The coordinates z,and z1 are unambiguously deterrespect to its fragments of length mined from the first equation (81, (n--i)s,i-2,-t,,using the value of t, . The remaining cox are also successively determined using (8). It turns out that ordinates of the vector #is reconstruction algorithm can be readily generalized to the case of an arbitrary kBd2. We shall consider the first equation of system (6): NOW let kBni2. 1--l+,

c I-I

z&y-t,.

The length of the column 2, is equal to C,*and Let 2, be the first column of the matrix A,. the coordinate I, is encountered u,,-ck$ times in it. All the remaining coordinates of the vector

are encountered

x=(ar,...,4

k>n12, u,,/vp=k/(n-k)>l. in accordance with 2% Next we eliminate in column

2,

, the coordinate

ates of the vector

x

a total of v, times in 2, where

v,=C,k-C,“r:=C,,“_,.

SlnCe

Whence if follows that 2~ is unambiguously determined from column the majority. all elements equal to 2, from column 2,. Of the elements remaining z+ is encountered

are encountered

u,p=C~~

times.

All the remaining

a total of v;:times in the reduced. column.

coordin-

,Nhere

192

v2=v,-_(I,:4~_z.

k>n/2,

Since

u,,lu,=:kl(n-k-l)>l.

It follows from this that the coordinate

determined in accordance with the majority from the reduced first 52 is also unambiguously column of matrix A.. By continuing the process of successively determining the variables x, according to the majority principle we find the values of x,,...,.r,,++, from the first column. However, system (6) is of triangular form and hence the values of the remaining coordinates are unambiguously determined from (6). The next theorem follows directly from this algorithm. Theorem

4.

k>n/2

When

system

(6) has just a single solution

in

(O,l}

numbers.

Corollary 3 When k>n/2 , the set E,*is complete. (see /2/), The lower bound for k for which the set E,'can be complete will be given next. Let V=E,‘. Let us assume that A=l/rq,ll is the matrix of the fragments due to the action of set V on vector a=(a,,...,a.). Each row of matrix A is a vector from EL . Then A can be defined with the help of a vector g(x): g(a)=(q,,...,q+), where the numeration (1,2,...,2*} cooresponds to the numeration of all the vectors from E* and the number q, is the number of times a vector occurs in matrix A. Since the number aFE* of rows in matrix A is equal to CmL, then

The number of solutions Theorem dition

5.

of this equation

In order

that the set

is

C,;+l)i_l.From this we have n

E,' should be complete, k c;Y&_r n

it is necessary

that the con-

22"

should be satisfied. Corollary

4.

The asymptotic

inequality

k> lgn lglgn is true. Below we shall attemp to elucidate to what extent information concerning the k-fragments of a vector is decisive in the sense of the possibility of establishing it in a typical a=E” To do this it is, as usual, necessary case. for us to establish certain facts of a probabilistic nature associated with the problem under consideration. Let a=E". We shall denote the set of different fragments of length k which are contained in the vector a by &(a) and the cardinal number of this set by E.. We shall call the set &j&(a) the k-neighbourhood of the vector a. We commence the account with a combinatorial lemma. Let (P"(P) be the number of different vectors from ,?Pcontaining a vector BEE*. Lemma

3.

The equation cp,, (a) =q,, (k) =C.9C:“+.

. . i-c.”

is true. We shall prove this equation by induction with respect to n. It can be directly verified for small values of R. Let p=(p,,..., ph). For each vector a=(a,,...,a), shall define a we function $@(a) as the maximum of the numbers of the coordinates of the vector such that ac=+. We shall write this fact briefly as Q)b(a)=i. We denote the set of points a=E” such that @sa, *p(a) =m by %,(P) . If d(p) is the set of all points from &"containing p, then &(B)nisl,(P)=0,

se(B)=,@%(B). By definition

(9)

i-i.

~=(a,,...,a~+~_,, aA+,,. . ..a.)EE”

Se,+,(p) is the set of points

#PI, i--l, 2,. , n-k-2; crr+r=p~xand aA+F+i (p,, , PA-,) E (af, .. ,a*+,-J. i:orn these conditions and the inductive hypothesis we have that Using (9) we obtain from this that cp,+,-*(k-l) when B=(~,,...,~M).

such that

1)

11-1

/-I

which was required. We shall denote Lemma

4.

n--l

t--*+1

,_I

,-”

v,(k-l\=z

q.(a)-z

the expectation

c

n-1

“-1

,-L--l

.-,

Il~*+,(P)II=:(pl+._,(al)=

n--l

C:+J-‘=~ c c,‘==c Cy”-cnk+. . .-kc.“,

of the quantity

i-*--L

&, defined

above by

M(b)

,

The equation M(f,)= zL (1 -$),

s.:-'-ECU' 1-o

(LO)

193

is true. Proof.

Let

Then

But, using Lemma

3, formula

(11) can be written

in the form

c !.oa-'P"(@)=C,,*f~~+'$-...+C,"=2"--sP'. aze" Whence M(f.)=2*(1-S:-‘/2n), which was required. Theorem 6. If k=o(n) asymptotically of 2'points. Proof.

a=E”

then almost all points

We shall estimate

the variance

of the random

have

a k-neighbourhood

quantity

consisting

f. from above. We have

that

Then

But (121,

Let

s&-'2-".

Then M{(&,)‘FM(E.)+

Next, by taking account

of

2y2L-l)

---+25+7i-‘)

=2*(i-e).

(10) and (12) , we obtain (13)

Since

k=o(n),

then lim e Cn)= lim +n-._ n-m

L-s -0.

(14)

Now by selecting an increasing A(n) and using (LO), (13), and (14), we obtain confirmation k-c(n) almost all points from EL have an identical of Theorem 6. Theorem 6 shows that, when k-neighbourhood and hence a knowledge of it evidently does not suffice for the complete reconstruction of the basic vector. REFERENCES 1. ZHURAVLEV YU.I., Extremal algorithms in mathematical models for problems of recognition and prediction, Dokl. Akad. Nauk SSSR, 231, 3, 761-763, 1976. from its fragments. II-I:Vychisl. Mat i vychisl. Tekhn. 2. KALASHNIK V.V., Word reconstruction (Computational Mathematics and Computational Techniques), No.4, 1973.

Translated

by E.J.S.