Let us recall the asymmetric IFSC introduced in [He and Nazer, 2016].
For a full-rank integer matrix A, assume that the combinations v1, . . . , vL have
been re-indexed (i.e., the rows of A) such that their effective variances are mono- tonically increasing (i.e., Ekv1k2 ≤ · · · ≤ EkvLk2). Furthermore, assume that the
sources are re-indexed (i.e., the columns of A, columns and rows of KSS as well as
the diagonal elements of D) such that the full-rank integer matrix A has full-rank sub-matrices As,[1:m], for m = 1, . . . , L.
Furthermore, define a permutation πF such that dπF(L) ≤ · · · ≤ dπF(1).
Codebook
Generate nested lattice codebooks C` , ΛF,` ∩ V(ΛC,`) with rates R` = 12log
θ
C,` θF,`
using nested lattices ΛC,L ⊆ · · · ⊆ ΛC,1 ⊆ ΛF,πF(1) ⊆ · · · ⊆ ΛF,πF(L) selected using
distortion level.
Remark 4. Note that for the symmetric rate case R, the monotonically increasing effective variances Ekv1k2 ≤ · · · ≤ EkvLk2 induces a monotonically increasing dis-
tortion levels d1 ≤ · · · ≤ dL where πF is the identity permutation in this case.
Compression
The compression part is similar to IFSC in Section 3.3.2, however with asymmetric fine lattice ΛF,` and coarse lattice ΛC,` such that the `th encoder obtains
t` = QΛF,`(s`+ u`)
λ` = [t`] mod ΛC,` (3.15)
where u` is a random dither that is independent of s` and uniformly distributed over
V(ΛF,`).
It is useful to write the ith combination as
vi†= a†i(T − U) , i = 1, . . . , L, (3.16)
where T = S + U + Q, Q , [q1 · · · qL]†, q` , −[s`+ u`] mod ΛF,`and is independent
of s` and uniformly distributed over V(ΛF,`) from the Crypto Lemma.
Algebraic Successive Decompression
Recall that in parallel decompression in (3.13), we found that by computing [A(Λ − U)] mod ΛC = [A([T] mod ΛC− U)] mod ΛC and using the distributive law, we were able to
recover [A(T − U)] mod ΛC, which can be written as [A (S + Q)] mod ΛC.
This was only possible since we were using a single coarse lattice ΛC. Unfortu-
nately, here at the mth decoding step, we take mod Λ
hold. Alternatively, at the mth decoding step, we can compute
[L(Λ − U)] mod ΛC,m= [L(T − U)] mod ΛC,m = [L(S + Q)] mod ΛC,m
where the first equality holds using the distributive law if and only if L is an upper triangular integer matrix. This is due to the fact that, the mth row of LΛ only
contains mod ΛC,`operations for ` ≥ m and for those values of ` we have ΛC,`⊆ ΛC,m
(i.e., ΛC,m is finer than all ”inner” modΛC,` operations).
Furthermore, the upper triangular property of L means that the mth combination
will only contain the source vectors sm, . . . , sL. Using previously decoded combina-
tions v1, . . . , vm−1(assuming correct decoding) as side information, we can obtain the
missing part of the mth combination (i.e., fill out the zero elements in the mth row of
the upper triangular matrix L).
Towards this end, let us define a strictly lower triangular integer matrix C where its mth row c†
m contains the coefficients of previously successfully decoded combinations
v1, . . . , vm−1, hence the strictly upper triangular condition on the matrix C.
Next, we discuss the details of our algebraic successive decompression. The de- coder first recovers the lattice codewords λ1, . . . , λL from the indices i1, . . . , iL, then
computes b v†m =h`†m(Λ − U) − c†mVb i mod ΛC,m (a) = h`†m(T − U) − c†mVb i mod ΛC,m (b) = `†m+ c†mA (T − U) mod ΛC,m (c) = `†m+ c†mA (S + Q) mod ΛC,m (d) = `† m+ c † mA mod p × (S + Q) mod ΛC,m (e)
= a†m mod p × (S + Q) mod ΛC,m
where bV , [bv1 · · · vbL]
†, c†
mV is available at the mb th decoding step since Cm,i = 0 for i ≥ m, (a) follows from the distributive law, (b) holds from assuming correct decoding for previous combinations (i.e., vbi = vi for i = 1, . . . , m − 1) and using
(3.16), (c) holds from T = S + U + Q, Q , [q1 · · · qL]†, qk , −[sk+ uk] mod ΛF,k
is independent of sk and uniformly distributed over V(ΛF,k) by the Crypto Lemma,
(d) holds from [Ordentlich et al., 2014, Theorem 2 (c)] and (e) holds from Lemma28 in Appendix B which states that for any full-rank integer matrix A with full-rank sub-matrices A[1:m]for m = 1, . . . , L, we can select an upper triangular integer matrix
L and a strictly lower triangular integer matrix C such that
[L + CA] mod p = [A] mod p. (3.18)
Furthermore, the mth estimated combination
b
vm† = v†m, m = 1, . . . , L, if a†m(S + Q) ∈ V (ΛC,m) which happens with high probability if
1
TEka
†
m(S + Q) k2 < θC,m, m = 1, . . . , L.
This can be guaranteed by setting θF,` = d` for ` = 1, . . . , L and θC,m =
a†m(KSS + D) am + , where T1ESS†
= KSS is the covariance matrix of the L
sources, 1
TE[QQ †
] , D = diag (d1, . . . , dL) and goes to zero as the blocklength goes
to infinity.
Finally, the decoder applies the inverse of A to obtain
b
S , A−1V = Ab −1V = S + Q.
Remark 5. The advantage from using L nested coarse lattices instead of a single one as in the parallel decompression, is that each coarse lattice ΛC,m should tolerate only
v1, . . . , vL (i.e., vm ∈ V(ΛC) w.h.p. for m = 1, . . . , L).
Theorem 2 ( [Bakoury and Nazer, 2017]). For a given distortion matrix D and covariance matrix KSS, the asymmetric achievable rates for IFSC with algebraic suc-
cessive decompression are
RSIFSC,`(KSS, D) = min A∈ZL×L 1 2log + a † `(KSS + D) a` d` ! , ` = 1, . . . , L (3.19)
where the minimization over all integer matrices A such that Rank(A[1:m]) = m for
m = 1, . . . , L and a†1(KSS+ D) a1 ≤ · · · ≤ a †
L(KSS+ D) aL.
Early, we have assumed that the sources have been re-indexed such that the matrix A has full-rank sub-matrices A[1:m] for m = 1, . . . , L. Later, in our work, we will need
to write the achievable rates in terms of the original source order.
Lemma 8. Define the permutation πSIFSC as the combinations re-ordering
that we did in the beginning to ensure that after re-ordering, we have
a†π
SIFSC(1)(KSS+ D) aπSIFSC(1) ≤ · · · ≤ a †
πSIFSC(L)(KSS+ D) aπSIFSC(L) and the permu-
tation πrank as the source re-ordering such that Rank(AπSIFSC([1:m]),πrank([1:m])) = m for
m = 1, . . . , L. The achievable rates in Theorem 2 in terms of the original source order can be written as
RSIFSC,πrank(`)(KSS, D) = min A∈ZL×L 1 2log + a † πSIFSC(`)(KSS+ D) aπSIFSC(`) dπrank(`) ! , ` = 1, . . . , L (3.20) where D = diag(d1, . . . , dL) and d` is the `th distortion level at the `th source.
Remark 6. Note that, to achieve the compression rates in (3.14) or (3.19), all sources need to know the covariance matrix KSS. On the other hand, to achieve the compres-
sion rates in (3.9), the `th source only needs to know its variance K SS,`,`.