Multiresolution quantization is a special case of multiple description quantization with r0+r1 = 1. However, recall that for multiresolution quantizers we assume that the second side quantizerq(2)restricted to a cell of q(1) is a nearest neighbor quantizer. Although this assumption is not compatible with our earlier assumption on the cell structure of multiple description quantizers, it allows, through similar methods, a simpler graph representation, and results in an algorithm with somewhat reduced complexity.
Similarly to the multiple description case, first we find a fine covering of the allowed multiresolution quantizers: Let QMRK ⊂ QMR denote the set of multiresolution scalar quantizers such that the cells of the first side quantizer (which is often referred to as the base quantizer) and the cells of the second side quantizer (the refinement quantizer) restricted to the cells of the base quantizer are right closed intervals with endpoints from Cˆ(K), all the reproduction points are also from ˆC(K), and belong to the corresponding interval cell.
Then, similarly to Lemma 2, it can be shown that for any quantizer q ∈ QMR there is a q0 ∈ QMRK such that
sup
x∈[0,1]
(dq0(x)−dq(x))≤3cρ/K. (52) (see the footnote in the proof of Lemma 2).
This result allows us to compete with best code inQMR by applying the coding scheme of Section 3 to the finite set of codes QMRK . There are two differences compared to the general multiple description case: (i) Since there is no loss on the first channel, it is enough to send the index of the chosen quantizerQ(k) only once in each block, requiring
h=
log|QMRK | logM1
≤ 1
logM1
log
K M1−1
K M1
K M1M2
(53) time instants (recall that the second side quantizer restricted to a cell of the base quantizer is assumed to be a nearest neighbor quantizer). (ii) The simpler structure ofQMRK allows a smaller graph representation. Indeed, the graph describing quantizers from QMRK can be constructed as follows: Define vertices (z, m1) for each z ∈ Cˆ(K)∪ {0,1} and m1 ∈ {0, . . . , M1}. The vertex (z, m1) corresponds to the case that the right endpoint of the m1st cell of q(1) is z (recall that the cells of q(1) are intervals). Now vertices (z1, m1) and (z2, m1 + 1) are connected via the following subgraph. The first vertex of the subgraph is (z1, z2, m1), and there are K(z2 −z1) edges going from (z1, m1) to (z1, z2, m1), each corresponding to a different possible code point of the cell from the set (z1, z2]∩Cˆ(K)with weight corresponding to the distortion of this cell in the kth block (if z1 = 0, then there are Kz2+ 1 edges, where the extra edge corresponds to the code point z1 = 0, which is a valid code point only in this case). Then (z1, z2, m1) and (z2, m1 + 1) are connected via a directed graph corresponding to an M2-level nearest neighbor quantizer from z1 to z2 as in Theorem 5, but the weights of the edges are multiplied by r0 (see Figure 3). The number of edges of such a subgraph is O(M2K2); hence the total number of edges of the constructed graph is O(M1M2K4).
Therefore, the complexity of the algorithm is slightly reduced compared to the general multiple description case, as it requires onlyO(M1M2K4n2/l2)+O(n3/l3)+O(M1M2K5n/l)+
O(n) computations.
From here, similarly to Theorem 6, we obtain the following result.
Theorem 7 Assume that m, n, l, K, M1, M2, h are positive integers such thatM1M2 ≤ K, l divides n, and h≤l, where h is defined in (53). Then for any 0< α <1, η >0, the normalized cumulative distortion of the above described coding scheme can be bounded for
z1, m1 z1, z1+K1, m1
z1+K1, m1+1
z1, z2, m1
z1,1, m1 1, m1
p p p
p p p p p
pp p
pp p
Subgraph for an M2-level nearest neighbor quantizer
z1 z2
f
ˆ x(1)m1+1
v
ˆ x(0)m1+1,1
v
ˆ x(0)m1+1,2
v
ˆ x(0)m1+1,3
v
ˆ x(0)m1+1,M2 Q:
p p p p p p p p p p p p
p p p p p p p
p p p p p p p
p p p p p p p
p p p
z2, m1+1
z1 z1+K2
0 1 2 3 M2
M2+1
Figure 3: A section of a multiple description scalar quantizer and the corresponding graph.
any sequence xn∈[0,1]n as E
"
1 n
Xn i=1
ρ(|xi−xˆi|)
#
−DQ∗MR,m,n(xn)
;
≤ hρ(1/2)
l + 1
ηnln
|QMRK |m+1 αm(1−α)n/l−m−1
+ηl2
8 + ml n + 7cρ
2K.
The algorithm can be implemented withO(M1M2K4n2/l2)+O(n3/l3)+O(M1M2K5n/l)+
O(n) computational complexity.
Remark. Optimizing the above bound as after Theorem 5, we obtain E
"
1 n
Xn i=1
ρ(|xi−xˆi|)
#
−DQ∗MR,m,n(xn)
≤ C1
logK l +C2
rlm n log n
lm + 7cρ
2K + ml n .
From here the best possible rate achievable isO((m/n)1/3log(n/m)), whenl =c1(n/m)1/3 and K =c2(n/m)1/3, which requires O(M1M2n8/3/m2/3) +O(mn2) computations.
On the other hand, if we set l = c1n3/4/m1/2 and K = c2n1/8/m1/4, then, assuming m = O(n1/6), the computational complexity of the algorithm is O(M1M2n), and the normalized distortion redundancy becomes O(m1/4√
logn/n1/8) (note that, similarly to the case of multiple description quantizers, omitting the factors associated with m from the definitions ofl and K results in a linear complexity regardless of the relation between m and n).
7 Conclusion
We presented a general scheme for limited-delay lossy coding of individual sequences.
For any finite class of limited-delay, finite-memory reference coders, our scheme performs asymptotically as well as the best “tracking” scheme which can change its coder a given number of times (each time choosing from the reference class) while encoding an input sequence of finite length. In order to implement the method, we devised an efficient algorithm for online prediction that tracks the best expert even when the number of experts is exponentially large, provided the experts have a certain additive structure.
The example of tracking the minimum-weight path in an acyclic graph was worked out in detail and used to construct efficient quantization schemes. In particular, we have constructed low-complexity schemes for tracking the best scalar quantizer, as well as the best multiple description or multiresolution quantizer (among the ones with interval cells).
For all these schemes, analyses of distortion redundancy and computational complexity were provided.
The focus of this work was on implementable sequential lossy coding schemes based on ideas rooted in the theory of sequential prediction for individual sequences. While the considered combined (i.e, tracking) scalar (network) quantization schemes form more powerful classes than the base class of scalar (network) quantizers, these schemes are still less general than one would desire in certain applications. For example, efficiently imple- mentable schemes for the class of coders with finite-state encoders and sliding-window (or finite-state) decoders would by desirable [2]. Also of interest would be the construction of linear-time scalar (network) quantization schemes with the same distortion redundancy rates as that of the more complex schemes presented here.
Appendix A
Proof of Theorem 1 The proof of Theorem 1 is done through a sequence of lemmas.
First observe that, denoting by Et the expectation taken with respect to Ut only, the sequence
Vt=`(yt,yˆt)−Et`(yt,yˆt) =`(yt,yˆt)−E[`(yt,yˆt)|Ut−1], t= 1, . . . , T
is a martingale difference with respect to U1, U2, . . . , UT such that with probability one
−E[`(yt,yˆt)|Ut−1]≤Vt ≤ −E[`(yt,yˆt)|Ut−1] +B.
Thus it suffices to bound the difference XT
t=1
Et`(yt,yˆt)−min
t,e L(P(T, m,t,e)) since
XT t=1
Vt=LT − XT
t=1
Et`(yt,yˆt)
and so by the Hoeffding-Azuma inequality [41] for sums of martingale differences, with probability at least 1−p,
LT ≤ XT
t=1
Et`(yt,yˆt) +B
rT ln(1/p) 2 and
E
" T X
t=1
Et`(yt,yˆt)
#
= XT
t=1
E`(yt,yˆt) .
Lemma 3 The cumulative loss of Algorithm 1 satisfies XT
t=1
Et`(yt,yˆt)≤ −1
ηlnwmT,i+T ηB2 8 for all i= 1, . . . , N.
Proof. The proof follows standard arguments, see, e.g., Cesa-Bianchi and Lugosi [42].
From the definition of Wt we have for all t lnWt+1
Wt
= ln PN
i=1wt,is e−η`(yt,ˆyt(i)) PN
i=1wst,i
!
= ln Ete−η`(yt,ˆyt)
= ln e−ηEt`(yt,ˆyt)Ete−η(`(yt,ˆyt)−Et`(yt,ˆyt))
= −ηEt`(yt,yˆt) + ln Ete−η(`(yt,ˆyt)−Et`(yt,ˆyt))
≤ −ηEt`(yt,yˆt) + η2B2 8 .
Note that the expectations are taken with respect to the random choice of ˆyt, that is, with respect to the randomizing variableUt, and the inequality holds by Hoeffding’s inequality [41] for the moment generation function of bounded random variables. Summing the above inequality for all t = 1, . . . , T, we obtain
lnWT+1
W1 ≤ −η XT
t=1
Et`(yt,yˆt) + T η2B2 8 . Since W1 = 1 andWT+1 ≥wT,im for any i, we have
lnwT,im ≤ −η XT
t=1
Et`(yt,yˆt) + T η2B2 8
which implies the statement of the lemma.
Lemma 4 For any 1≤t ≤t0 ≤T and any i= 1, . . . , N we have wtm0,i
wt,is ≥e−ηL([t,t0],i)(1−α)t−t0 where L([t, t0], i) =Pt0
τ=t`(yτ,yˆτ(i)).
Proof. The proof is a straightforward modification of the one in [21]. From the definitions of wmt,i and wt+1,is (see equations (2) and (3)) it is clear that for any τ ≥1
wsτ+1,i = αWτ+1
+ (1−α)wmτ,i ≥(1−α)e−η`(yτ,ˆyτ(i))wτ,is .
Applying this equation iteratively forτ =t, t+ 1, . . . , t0−1, and (2) for τ =t0, we obtain wmt0,i ≥e−η`(yt0,ˆy(i)t0 )
tY0−1 τ=t
(1−α)e−η`(yτ,ˆy(i)τ )
wt,is =e−ηL([t,t0],i)(1−α)t−t0wst,i
which implies the statement of the lemma.
Lemma 5 For any t≥1 and 1≤i, j ≤N, we have wst+1,i
wmt,j ≥ α N.
Proof. By the definition of wt+1,is we have wst+1,i ≥ αWt+1
N = α
N XN
l=1
wmt,l≥ αwmt,j N .
This completes the proof of the lemma.
Proof of Theorem 1. LetP(T, m,t,e) be an arbitrary partition. Then by Lemma 3, the cumulative loss of Algorithm 1 can be bounded as
XT t=1
Et`(yt,yˆt)≤ −1
η lnwmT,em+T ηB2
8 (A.1)
(recall that em denotes the expert used in the last segment of the partition). Now wT,em m can be rewritten in the form of the following telescoping product
wT,em m =wts0+1,e0 wtm1,e0 wts0+1,e0
Ym i=1
wsti+1,ei wmti,ei−1
wmti+1,ei wsti+1,ei
! .
Therefore, applying Lemmas 4 and 5, we have wmT,em ≥ wst0+1,e0α
N
mYm i=0
e−ηL((ti,ti+1],ei)(1−α)ti+1−ti−1
= 1
Ne−ηL(P(T,m,t,e))(1−α)T−m−1α N
m
.
Substituting this bound in (A.1) proves (4) and (5), the first statements of the theorem.
To prove the second part, let H(p) = −plnp−(1−p) ln(1−p) and D(pkq) = plnpq+ (1−p) ln 1−p1−q. Optimizing the value of η in (4) gives the following:
XT t=1
Et`(yt,yˆt)−min
t,e L(P(T, m,t,e))
≤ B sT
2
(m+ 1) lnN+mln 1
α + (T −m−1) ln 1 1−α
= B rT
2 ((m+ 1) lnN + (T −1)(Db(α∗kα)−Hb(α∗))) where α∗ = Tm−1. For α=α∗ the bound becomes
XT t=1
Et`(yt,yˆt)−min
t,e L(P(T, m,t,e))
≤ BT1/2
√2 s
(m+ 1) lnN +mlnT −1
m + (T −m−1) ln
1 + m
T −m−1
≤ T1/2 B
√2 r
(m+ 1) lnN+mlnT −1
m +m
where we used the fact that ln(1 +x)≤x for all x >−1.
Appendix B
Proof of Lemma 2 In view of (32), the proof would be simple if only nearest neighbor quantizers were allowed as side quantizers. As this is not the case, a more involved construction is needed.
For i= 1,2, let 0 =t(i)0 < t(i)1 <· · ·< t(i)Mi−1 < t(i)Mi = 1 denote the decision thresholds of q(i), the ith side quantizer of q (that is, the cells of q(i) are intervals with endpoints t(i)j−1 and t(i)j , j = 1, . . . , Mi). We will construct a sequence of two-description quantizers qj, j = 0, . . . , K, such that all the thresholds t of the side quantizers qj(i) that satisfy t ≤ j/K are of the form t = k/K for some integer k, the corresponding cells with right end points t are right closed intervals, and qj+1 and qj differ only in the interval (j/K,(j+ 1)/K].
Let q0 = q, which clearly satisfies that all thresholds t ≤ 0 are of the form 0/K.
Assume that we have already constructed qj with the desired property for some j ≥ 0.
From qj we construct qj+1 by modifying the thresholds of the side quantizers of qj in the interval (j/K,(j+ 1)/K]. Therefore, for all x6∈(j/K,(j+ 1)/K],
dqj+1(x)−dqj(x) = 0. (B.1)
Furthermore, if qj has no threshold in the interval (j/K,(j+ 1)/K], then letqj+1 =qj
(so obviously dqj =dqj+1).
Ifqj has exactly one threshold in (j/K,(j+ 1)/K), then without loss of generality we can assume that this thresholdt belongs to the first side quantizerqj(1). Letqlef t andqright
be 2-description quantizers obtained from qj by replacing the threshold t with j/K and (j+ 1)/K, respectively (such that the new threshold is quantized to a smaller code point;
that is, the corresponding cell is closed from the right), and qj+1 will be the one with smaller guaranteed worst case distortion. Let c(1)1 , c(1)2 denote the code points of the side quantizer qj(1) corresponding to the cells ending and beginning at t, respectively, and let c(0)1 , c(0)2 denote the corresponding code points of the central quantizer qj(0). Now clearly, if x≤j/K orx > t, then qj(x) = qlef t(x). For x∈(j/K, t), we have
dqlef t(x)−dqj(x)
= r0
ρ(|x−c(0)2 |)−ρ(|x−c(0)1 |) +r1
ρ(|x−c(1)2 |)−ρ(|x−c(1)1 |)
≤ r0
ρ(|t−c(0)2 |)−ρ(|t−c(0)1 |) + 2cρ(t−j/K) +r1
ρ(|t−c(1)2 |)−ρ(|t−c(1)1 |) + 2cρ(t−j/K)
≤ r0
ρ(|t−c(0)2 |)−ρ(|t−c(0)1 |) +r1
ρ(|t−c(1)2 |)−ρ(|t−c(1)1 |)
+ 2cρ/K (B.2) and similarly, for x < t orx >(j+ 1)/K, qj(x) = qright(x), and forx∈(t,(j+ 1)/K),
dqright(x)−dqj(x)
= r0
ρ(|x−c(0)1 |)−ρ(|x−c(0)2 |) +r1
ρ(|x−c(1)1 |)−ρ(|x−c(1)2 |)
≤ r0
ρ(|t−c(0)1 |)−ρ(|t−c(0)2 |) +r1
ρ(|t−c(1)1 |)−ρ(|t−c(1)2 |)
+ 2cρ/K. (B.3) It is easy to see that for x = t, either qj(x) = qlef t(x) (resp. qj(x) = qright(x)), or the bound (B.2) (resp. (B.3)) holds. Now let qj+1 = qlef t if the bound (B.2) is smaller than (B.3), and letqj+1 =qright otherwise. Then, as the first two terms in (B.2) and (B.3) are the negative of each other,
dqj+1(x)−dqj(x)≤2cρ/K. (B.4) for all x∈(j/K,(j+ 1)/K].
If one of the side quantizers of qj has at least one cell that is contained in the interval (j/K,(j+ 1)/K], then in qj+1, all such cells are merged and enlarged into the larger cell (j/K,(j + 1)/K] with reconstruction point (j+ 1/2)/K (consequently, the neighboring cells may shrink). If the other side quantizer has no threshold in (j/K,(j+ 1)/K], then no other modification is necessary. If it has exactly one threshold, then similarly to (B.2) and (B.3), that threshold can be moved either to j/K or to (j + 1)/K. Finally, if the other side quantizer also has at least two thresholds in the interval (j/K,(j+ 1)/K], then merging all cells in this interval as for the other side quantizer results in a quantizerqj+1
for which (j/K,(j+ 1)/K] is a cell of both side quantizers, and consequently, it is also a cell of the central quantizer. In all cases, the bound (B.4) holds for the distortion ofqj+1. The last case to be considered is when both side quantizers have exactly one threshold in (j/K,(j+ 1)/K]. Lett1 ≤t2 denote these thresholds. Ift1 =t2 and there is no central
partition cell {t1}, then, similarly to (B.2) and (B.3), it can be shown that replacing the common threshold of the side quantizers with j/K or (j + 1)/K results in a maximum increase in distortion of 3cρ/K. Note that the factor 2 in (B.4) is replaced with 3 because here all three quantizers (both side quantizers and the central quantizer) change.2 If t1 < t2, then similarly to (B.4), it can be shown that t1 can be replaced with j/K or t2
at the price of an increase of at most 2cρ/K in the pointwise distortion (such that the new central quantizer has no one-point cell {t2}). Then, the thresholdt2 (which may be a single or a common threshold) can be quantized into ˆC(K) as before on the price of further increasing the pointwise distortion by at most 3cρ/K. Finally, ift1 =t2 and the central partition has a one-point cell {t1}, then similarly to the case t1 < t2, first we can replace this cell by the empty cell or by (j/K, t2], and then proceed as in the previous case. Based on the above, we have, for all x∈(j/K,(j+ 1)/K],
dqj+1(x)−dqj(x)≤5cρ/K. (B.5) From equations (B.1), (B.4), and (B.5) we can see that for all x∈[0,1],
dqK(x)−dq(x)≤5cρ/K.
Now let q0 ∈ QMDK be a quantizer obtained from qK by replacing its code points by the corresponding closest points in the set ˆC(K) that also belong to the corresponding quantization cells. Then
dq0(x)−dq(x)≤ 5cρ
K + cρ
K = 6cρ
K
for all x∈[0,1], as desired.
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