Soft-Covering ExponentPaul Cuff and Semih Yagli

Soft Covering

- Opposite of Reliable Decoding - What happens when rate is too high?

Noise

X1

Y1

Noise

X2

Y2

Noise

X3

Y3

Noise

X4

Y4

i.i.d.

i.i.d.

Codebook

i.i.d.?

Output Distribution

PY

PY|C

x(1) x(2) x(3)x(4)x(5)

Gaussian Example

Gaussian Example

Soft Covering (origin)

Theorem 6.3 of Wyner’s Common Information Paper

R > I(X;Y) produces the phenomenon

Various Names

Covering [Ahlswede, Winter, Wilde]

Sampling Lemma [Winter]

Resolvability [Han, Verdu]

Cloud Mixing

Applications

Security and Privacy

e.g. wiretap channel

Encoder analysis

Common Information

Main Result

Previous lower bounds

Notable bounds by Hayashi, Han, Verdu, Cuff

Best previous:

TV vs KL-divergence

Inspired by [Parizi, Telatar, Merhav 17]

Solved exponent for KL-divergence

Features of TV

Accepted standard for cryptography

Tractable multi-stage encoder analysis

Stronger concentration result

New Features of TV Proof

“Typical set” approach is not optimal

Poisson codebook size used in converse

(Taylor expansion not possible for absolute value)

Equivalence

Carefully swap min’s and max’s

Massage

[R�D(QXY kPXQY )]+ = max�2[0,1]

�(R�D(QXY kPXQY ))<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Proof of Error Upper Bound

Illustration of codebook approximation

Ynyn

Conditional Type

0 0 0 0 1 0 0 00 1 0 1 1 1 0 11 1 1 1 1 0 0 0

yn

xn same conditional type

Polynomial Number of Types

Length-n binary types

fraction of 1’s0 1/n 2/n 1

Length-n ternary types

fraction of 1’s

fraction of 2’s

1

1

Key Quantities

Bound each type in one of two ways

Proof of Error Lower Bound

Poisson Trick

Let the number of codewords M be Poisson distributed

Take care of this assumption at the end

Another Look

Need independence for different types

A Simple Lemma{Xi} independent

E[Xi]=0

E�����X

i

Xi

����� � maxi

E |Xi|<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Initial Details

Poisson to Fixed Codebook

Codebook Size (M)

codebook size PMFexponentially small

(still too big to be bad)

μ