arXiv subjects

Cheuk Ting Li

Publications and source records attributed to Cheuk Ting Li.

At least 37 records · Page 2Linked to original sources

arXiv2024

Variable-Length Secret Key Agreement via Random Stopping Time

We consider a key agreement setting where two parties observe correlated random sources, and want to agree on a secret key via public discussions. In order to allow the key length to adapt to the realizations of the random sources, we allow the key to be of variable length, subject to a novel variable-length version of the uniformity constraint based on random stopping time. We propose simple, computationally efficient key agreement schemes under the new constraint. The proposed scheme can be considered as the key agreement analogue of variable-length source coding via Huffman coding, and the Knuth-Yao random number generator.

cs.IT↗

arXiv2024

A Pair of Bayesian Network Structures has Undecidable Conditional Independencies

Given a Bayesian network structure (directed acyclic graph), the celebrated d-separation algorithm efficiently determines whether the network structure implies a given conditional independence relation. We show that this changes drastically when we consider two Bayesian network structures instead. It is undecidable to determine whether two given network structures imply a given conditional independency, that is, whether every collection of random variables satisfying both network structures must also satisfy the conditional independency. Although the approximate combination of two Bayesian networks is a well-studied topic, our result shows that it is fundamentally impossible to accurately combine the knowledge of two Bayesian network structures, in the sense that no algorithm can tell what conditional independencies are implied by the two network structures. We can also explicitly construct two Bayesian network structures, such that whether they imply a certain conditional independency is unprovable in the ZFC set theory, assuming ZFC is consistent.

cs.CC↗

arXiv2024

A Characterization of Entropy as a Universal Monoidal Natural Transformation

We show that the essential properties of entropy (monotonicity, additivity and subadditivity) are consequences of entropy being a monoidal natural transformation from the under category functor $-/\mathsf{LProb}_ρ$ (where $\mathsf{LProb}_ρ$ is category of $ρ$-th-power-summable probability distributions, $0<ρ<1$) to $Δ_{\mathbb{R}}$. Moreover, the Shannon entropy can be characterized as the universal monoidal natural transformation from $-/\mathsf{LProb}_ρ$ to the category of integrally closed partially ordered abelian groups (a reflective subcategory of the lax-slice 2-category over $\mathsf{MonCat}_{\ell}$ in the 2-category of monoidal categories), providing a succinct characterization of Shannon entropy as a reflection arrow. We can likewise define entropy for every monoidal category with a monoidal structure on its under categories (e.g. the category of finite abelian groups, the category of finite inhabited sets, the category of finite dimensional vector spaces, and the augmented simplex category) via the reflection arrow. This implies that all these entropies over different categories are components of a single natural transformation (the unit of the idempotent monad), allowing us to connect these entropies in a natural manner. We also provide a universal characterization of the conditional Shannon entropy based on the chain rule which, unlike the characterization of information loss by Baez, Fritz and Leinster, does not require any continuity assumption.

math.CT↗

arXiv2024

Pointwise Redundancy in One-Shot Lossy Compression via Poisson Functional Representation

We study different notions of pointwise redundancy in variable-length lossy source coding. We present a construction of one-shot variable-length lossy source coding schemes using the Poisson functional representation, and give bounds on its pointwise redundancy for various definitions of pointwise redundancy. This allows us to describe the distribution of the encoding length in a precise manner. We also generalize the result to the one-shot lossy Gray-Wyner system.

cs.IT↗

arXiv2024

Vector Quantization with Error Uniformly Distributed over an Arbitrary Set

For uniform scalar quantization, the error distribution is approximately a uniform distribution over an interval (which is also a 1-dimensional ball). Nevertheless, for lattice vector quantization, the error distribution is uniform not over a ball, but over the basic cell of the quantization lattice. In this paper, we construct vector quantizers with periodic properties, where the error is uniformly distributed over the n-ball, or any other prescribed set. We then prove upper and lower bounds on the entropy of the quantized signals. We also discuss how our construction can be applied to give a randomized quantization scheme with a nonuniform error distribution.

cs.IT↗

arXiv2023

Compression with Exact Error Distribution for Federated Learning

Compression schemes have been extensively used in Federated Learning (FL) to reduce the communication cost of distributed learning. While most approaches rely on a bounded variance assumption of the noise produced by the compressor, this paper investigates the use of compression and aggregation schemes that produce a specific error distribution, e.g., Gaussian or Laplace, on the aggregated data. We present and analyze different aggregation schemes based on layered quantizers achieving exact error distribution. We provide different methods to leverage the proposed compression schemes to obtain compression-for-free in differential privacy applications. Our general compression methods can recover and improve standard FL schemes with Gaussian perturbations such as Langevin dynamics and randomized smoothing.

cs.LG↗

arXiv2023

Communication-Efficient Laplace Mechanism for Differential Privacy via Random Quantization

We propose the first method that realizes the Laplace mechanism exactly (i.e., a Laplace noise is added to the data) that requires only a finite amount of communication (whereas the original Laplace mechanism requires the transmission of a real number) while guaranteeing privacy against the server and database. Our mechanism can serve as a drop-in replacement for local or centralized differential privacy applications where the Laplace mechanism is used. Our mechanism is constructed using a random quantization technique. Unlike the simple and prevalent Laplace-mechanism-then-quantize approach, the quantization in our mechanism does not result in any distortion or degradation of utility. Unlike existing dithered quantization and channel simulation schemes for simulating additive Laplacian noise, our mechanism guarantees privacy not only against the database and downstream, but also against the honest but curious server which attempts to decode the data using the dither signals.

cs.CR↗

arXiv2023

A Poisson Decomposition for Information and the Information-Event Diagram

Information diagram and the I-measure are useful mnemonics where random variables are treated as sets, and entropy and mutual information are treated as a signed measure. Although the I-measure has been successful in machine proofs of entropy inequalities, the theoretical underpinning of the ``random variables as sets'' analogy has been unclear until the recent works on mappings from random variables to sets by Ellerman (recovering order-$2$ Tsallis entropy over general probability space), and Down and Mediano (recovering Shannon entropy over discrete probability space). We generalize these constructions by designing a mapping which recovers the Shannon entropy (and the information density) over general probability space. Moreover, it has an intuitive interpretation based on the arrival time in a Poisson process, allowing us to understand the union, intersection and difference between (sets corresponding to) random variables and events. Cross entropy, KL divergence, and conditional entropy given an event, can be obtained as set intersections. We propose a generalization of the information diagram that also includes events, and demonstrate its usage by a diagrammatic proof of Fano's inequality.

cs.IT↗

arXiv2023

Weighted Parity-Check Codes for Channels with State and Asymmetric Channels

In this paper, we introduce a new class of codes, called weighted parity-check codes, where each parity-check bit has a weight that indicates its likelihood to be one (instead of fixing each parity-check bit to be zero). It is applicable to a wide range of settings, e.g. asymmetric channels, channels with state and/or cost constraints, and the Wyner-Ziv problem, and can provably achieve the capacity. For the channels with state (Gelfand-Pinsker) setting, the proposed coding scheme has two advantages compared to the nested linear code. First, it achieves the capacity of any channel with state (e.g. asymmetric channels). Second, simulation results show that the proposed code achieves a smaller error rate compared to the nested linear code. We also discuss a sparse construction where the belief propagation algorithm can be applied to improve the coding efficiency.

cs.IT↗

arXiv2023

Unconditionally Secure Access Control Encryption

Access control encryption (ACE) enforces, through a sanitizer as the mediator, that only legitimate sender-receiver pairs can communicate, without the sanitizer knowing the communication metadata, including its sender and recipient identity, the policy over them, and the underlying plaintext. Any illegitimate transmission is indistinguishable from pure noise. Existing works focused on computational security and require trapdoor functions and possibly other heavyweight primitives. We present the first ACE scheme with information-theoretic security (unconditionally against unbounded adversaries). Our novel randomization techniques over matrices realize sanitization (traditionally via homomorphism over a fixed randomness space) such that the secret message in the hidden message subspace remains intact if and only if there is no illegitimate transmission.

cs.CR↗

arXiv2022

Species Abundance Distribution and Species Accumulation Curve: A General Framework and Results

We build a general framework which establishes a one-to-one correspondence between species abundance distribution (SAD) and species accumulation curve (SAC). The appearance rates of the species and the appearance times of individuals of each species are modeled as Poisson processes. The number of species can be finite or infinite. Hill numbers are extended to the framework. We introduce a linear derivative ratio family of models, $\mathrm{LDR}_1$, of which the ratio of the first and the second derivatives of the expected SAC is a linear function. A D1/D2 plot is proposed to detect this linear pattern in the data. By extrapolation of the curve in the D1/D2 plot, a species richness estimator that extends Chao1 estimator is introduced. The SAD of $\mathrm{LDR}_1$ is the Engen's extended negative binomial distribution, and the SAC encompasses several popular parametric forms including the power law. Family $\mathrm{LDR}_1$ is extended in two ways: $\mathrm{LDR}_2$ which allows species with zero detection probability, and $\mathrm{RDR}_1$ where the derivative ratio is a rational function. Real data are analyzed to demonstrate the proposed methods. We also consider the scenario where we record only a few leading appearance times of each species. We show how maximum likelihood inference can be performed when only the empirical SAC is observed, and elucidate its advantages over the traditional curve-fitting method.

stat.AP↗

arXiv2022

An Automated Theorem Proving Framework for Information-Theoretic Results

We present a versatile automated theorem proving framework capable of automated discovery, simplification and proofs of inner and outer bounds in network information theory, deduction of properties of information-theoretic quantities (e.g. Wyner and Gács-Körner common information), and discovery of non-Shannon-type inequalities, under a unified framework. Our implementation successfully generated proofs for 32 out of 56 theorems in Chapters 1-14 of the book Network Information Theory by El Gamal and Kim. Our framework is based on the concept of existential information inequalities, which provides an axiomatic framework for a wide range of problems in information theory.

cs.IT↗

arXiv2022

Undecidability of Network Coding, Conditional Information Inequalities, and Conditional Independence Implication

We resolve three long-standing open problems, namely the (algorithmic) decidability of network coding, the decidability of conditional information inequalities, and the decidability of conditional independence implication among random variables, by showing that these problems are undecidable. The proof utilizes a construction inspired by Herrmann's arguments on embedded multivalued database dependencies, a network studied by Dougherty, Freiling and Zeger, together with a novel construction to represent group automorphisms on top of the network.

cs.IT↗

arXiv2022

The Undecidability of Conditional Affine Information Inequalities and Conditional Independence Implication with a Binary Constraint

We establish the undecidability of conditional affine information inequalities, the undecidability of the conditional independence implication problem with a constraint that one random variable is binary, and the undecidability of the problem of deciding whether the intersection of the entropic region and a given affine subspace is empty. This is a step towards the conjecture on the undecidability of conditional independence implication. The undecidability is proved via a reduction from the periodic tiling problem (a variant of the domino problem). Hence, one can construct examples of the aforementioned problems that are independent of ZFC (assuming ZFC is consistent).

cs.IT↗

arXiv2022

The Undecidability of Network Coding with some Fixed-Size Messages and Edges

We consider a network coding setting where some of the messages and edges have fixed alphabet sizes, that do not change when we increase the common alphabet size of the rest of the messages and edges. We prove that the problem of deciding whether such network admits a coding scheme is undecidable. This can be considered as a partial solution to the conjecture that network coding (without fixed-size messages/edges) is undecidable. The proof, which makes heavy use of analogies with digital circuits, is essentially constructing a digital circuit of logic gates and flip-flops within a network coding model that is capable of simulating an arbitrary Turing machine.

cs.IT↗

arXiv2022

Arithmetic Network Coding for Secret Sum Computation

We consider a network coding problem where the destination wants to recover the sum of the signals (Gaussian random variables or random finite field elements) at all the source nodes, but the sum must be kept secret from an eavesdropper that can wiretap on a subset of edges. This setting arises naturally in sensor networks and federated learning, where the secrecy of the sum of the signals (e.g. weights, gradients) may be desired. While the case for finite field can be solved, the case for Gaussian random variables is surprisingly difficult. We give a simple conjecture on the necessary and sufficient condition under which such secret computation is possible for the Gaussian case, and prove the conjecture when the number of wiretapped edges is at most 2.

cs.IT↗

arXiv2021

Multiple-Output Channel Simulation and Lossy Compression of Probability Distributions

We consider a variant of the channel simulation problem with a single input and multiple outputs, where Alice observes a probability distribution $P$ from a set of prescribed probability distributions $\mathbb{\mathcal{P}}$, and sends a prefix-free codeword $W$ to Bob to allow him to generate $n$ i.i.d. random variables $X_{1},X_{2,}...,X_{n}$ which follow the distribution $P$. This can also be regarded as a lossy compression setting for probability distributions. This paper describes encoding schemes for three cases of $P$: $P$ is a distribution over positive integers, $P$ is a continuous distribution over $[0,1]$ with a non-increasing pdf, and $P$ is a continuous distribution over $[0,\infty)$ with a non-increasing pdf. We show that the growth rate of the expected codeword length is sub-linear in $n$ when a power law bound is satisfied. An application of multiple-outputs channel simulation is the compression of probability distributions.

cs.IT↗

arXiv2021

First-Order Theory of Probabilistic Independence and Single-Letter Characterizations of Capacity Regions

We consider the first-order theory of random variables with the probabilistic independence relation, which concerns statements consisting of random variables, the probabilistic independence symbol, logical operators, and existential and universal quantifiers. Although probabilistic independence is the only non-logical relation included, this theory is surprisingly expressive, and is able to interpret the true first-order arithmetic over natural numbers (and hence is undecidable). We also construct a single-letter characterization of the capacity region for a general class of multiuser coding settings (including broadcast channel, interference channel and relay channel), using a first-order formula. We then introduce the linear entropy hierarchy to classify single-letter characterizations according to their complexity.

cs.IT↗