arXiv subjects
Publications and source records attributed to Jingbo Liu.
In this paper, we prove that $\boldsymbolζ$ cannot be a solution to any nontrivial algebraic differential equation whose coefficients are polynomials in $\boldsymbolΓ,\boldsymbolΓ^{(n)}$ and $\boldsymbolΓ^{(\ell n)}$ over the ring of polynomials in $\mathbf{C}$, where $\ell,n\geq1$ are positive integers.
A strong converse shows that no procedure can beat the asymptotic (as blocklength $n\to\infty$) fundamental limit of a given information-theoretic problem for any fixed error probability. A second-order converse strengthens this conclusion by showing that the asymptotic fundamental limit cannot be exceeded by more than $O(\tfrac{1}{\sqrt{n}})$. While strong converses are achieved in a broad range of information-theoretic problems by virtue of the "blowing-up method"---a powerful methodology due to Ahlswede, Gács and Körner (1976) based on concentration of measure---this method is fundamentally unable to attain second-order converses and is restricted to finite-alphabet settings. Capitalizing on reverse hypercontractivity of Markov semigroups and functional inequalities, this paper develops the "smoothing-out" method, an alternative to the blowing-up approach that does not rely on finite alphabets and that leads to second-order converses in a variety of information-theoretic problems that were out of reach of previous methods.
A new proof is given for the fact that centered gaussian functions saturate the Euclidean forward-reverse Brascamp-Lieb inequalities, extending the Brascamp-Lieb and Barthe theorems. A duality principle for best constants is also developed, which generalizes the fact that the best constants in the Brascamp-Lieb and Barthe inequalities are equal. Finally, as the title hints, the main results concerning finiteness, structure and gaussian-extremizability for the Brascamp-Lieb inequality due to Bennett, Carbery, Christ and Tao are generalized to the setting of the forward-reverse Brascamp-Lieb inequality.
The analysis of Belief Propagation and other algorithms for the {\em reconstruction problem} plays a key role in the analysis of community detection in inference on graphs, phylogenetic reconstruction in bioinformatics, and the cavity method in statistical physics. We prove a conjecture of Evans, Kenyon, Peres, and Schulman (2000) which states that any bounded memory message passing algorithm is statistically much weaker than Belief Propagation for the reconstruction problem. More formally, any recursive algorithm with bounded memory for the reconstruction problem on the trees with the binary symmetric channel has a phase transition strictly below the Belief Propagation threshold, also known as the Kesten-Stigum bound. The proof combines in novel fashion tools from recursive reconstruction, information theory, and optimal transport, and also establishes an asymptotic normality result for BP and other message-passing algorithms near the critical threshold.
We characterize the communication complexity of the following distributed estimation problem. Alice and Bob observe infinitely many iid copies of $ρ$-correlated unit-variance (Gaussian or $\pm1$ binary) random variables, with unknown $ρ\in[-1,1]$. By interactively exchanging $k$ bits, Bob wants to produce an estimate $\hatρ$ of $ρ$. We show that the best possible performance (optimized over interaction protocol $Π$ and estimator $\hat ρ$) satisfies $\inf_{Π\hatρ}\sup_ρ\mathbb{E} [|ρ-\hatρ|^2] = \tfrac{1}{k} (\frac{1}{2 \ln 2} + o(1))$. Curiously, the number of samples in our achievability scheme is exponential in $k$; by contrast, a naive scheme exchanging $k$ samples achieves the same $Ω(1/k)$ rate but with a suboptimal prefactor. Our protocol achieving optimal performance is one-way (non-interactive). We also prove the $Ω(1/k)$ bound even when $ρ$ is restricted to any small open sub-interval of $[-1,1]$ (i.e. a local minimax lower bound). Our proof techniques rely on symmetric strong data-processing inequalities and various tensorization techniques from information-theoretic interactive common-randomness extraction. Our results also imply an $Ω(n)$ lower bound on the information complexity of the Gap-Hamming problem, for which we show a direct information-theoretic proof.
A fundamental tool in network information theory is the covering lemma, which lower bounds the probability that there exists a pair of random variables, among a give number of independently generated candidates, falling within a given set. We use a weighted sum trick and Talagrand's concentration inequality to prove new mutual covering bounds. We identify two interesting applications: 1) When the probability of the set under the given joint distribution is bounded away from 0 and 1, the covering probability converges to 1 \emph{doubly} exponentially fast in the blocklength, which implies that the covering lemma does not induce penalties on the error exponents in the applications to coding theorems. 2) Using Hall's marriage lemma, we show that the maximum difference between the probability of the set under the joint distribution and the covering probability equals half the minimum total variation distance between the joint distribution and any distribution that can be simulated by selecting a pair from the candidates. Thus we use the mutual covering bound to derive the exact error exponent in the joint distribution simulation problem. In both applications, the determination of the exact exponential (or doubly exponential) behavior relies crucially on the sharp concentration inequality used in the proof of the mutual covering lemma.
We prove some uniqueness results for the Riemann zeta-function and the Euler gamma-function by virtue of shared values using the value distribution theory.
In this paper we study universal quadratic polynomials which arise as sums of polygonal numbers. Specifically, we determine an asymptotic upper bound (as a function of $m$) on the size of the set $S_m\subset\mathbb{N}$ such that if a sum of $m$-gonal numbers represents $S_m$, then it represents $\mathbb{N}$.
In this note, we will prove that $\mathboldζ$ and $\mathboldΓ$ can not satisfy any differential equation generated through a family of functions continuous in $\mathboldζ$ with polynomials in $\mathboldΓ$.
The primitive relay channel, introduced by Cover in 1987, is the simplest single-source single-destination network model that captures some of the most essential features and challenges of relaying in wireless networks. Recently, Wu and Ozgur developed upper bounds on the capacity of this channel that are tighter than the cutset bound. In this paper, we recover, generalize and improve their upper bounds with simpler proofs that rely on a converse technique recently introduced by Liu, van Handel and Verdú that builds on reverse hypercontractivity. To our knowledge, this is the first application of reverse hypercontractivity for proving first-order converses in network information theory.
This paper introduces a new converse machinery for a challenging class of distributed source-type problems (e.g.\ distributed source coding, common randomness generation, or hypothesis testing with communication constraints), through the example of the Wyner-Ahlswede-Körner network. Using the functional-entropic duality and the reverse hypercontractivity of the transposition semigroup, we lower bound the error probability for each joint type. Then by averaging the error probability over types, we lower bound the $c$-dispersion (which characterizes the second-order behavior of the weighted sum of the rates of the two compressors when a nonvanishing error probability is small) as the variance of the gradient of $\inf_{P_{U|X}}\{cH(Y|U)+I(U;X)\}$ with respect to $Q_{XY}$, the per-letter side information and source distribution. In comparison, using standard achievability arguments based on the method of types, we upper-bound the $c$-dispersion as the variance of $c\imath_{Y|U}(Y|U)+\imath_{U;X}(U;X)$, which improves the existing upper bounds but has a gap to the aforementioned lower bound.
We introduce an inequality which may be viewed as a generalization of both the Brascamp-Lieb inequality and its reverse (Barthe's inequality), and prove its information-theoretic (i.e.\ entropic) formulation. This result leads to a unified approach to functional inequalities such as the variational formula of Rényi entropy, hypercontractivity and its reverse, strong data processing inequalities, and transportation-cost inequalities, whose utility in the proofs of various coding theorems has gained growing popularity recently. We show that our information-theoretic setting is convenient for proving properties such as data processing, tensorization, convexity (Riesz-Thorin interpolation) and Gaussian optimality. In particular, we elaborate on a "doubling trick" used by Lieb and Geng-Nair to prove several results on Gaussian optimality. Several applications are discussed, including a generalization of the Brascamp-Lieb inequality involving Gaussian random transformations, the determination of Wyner's common information of vector Gaussian sources, and the achievable rate region of certain key generation problems in the case of vector Gaussian sources.
We discuss equivalence conditions on the non-existence of non-trivial meromorphic solution to the Fermat Diophantine equations $f^m(z)+g^n(z)=1$ with integers $m,n\geq2$, from which other approaches to prove little Picard theorem are provided.
The conventional channel resolvability refers to the minimum rate needed for an input process to approximate the channel output distribution in total variation distance. In this paper we study $E_γ$-resolvability, in which total variation is replaced by the more general $E_γ$ distance. A general one-shot achievability bound for the precision of such an approximation is developed. Let $Q_{\sf X|U}$ be a random transformation, $n$ be an integer, and $E\in(0,+\infty)$. We show that in the asymptotic setting where $γ=\exp(nE)$, a (nonnegative) randomness rate above $\inf_{Q_{\sf U}: D(Q_{\sf X}\|{π}_{\sf X})\le E} \{D(Q_{\sf X}\|{π}_{\sf X})+I(Q_{\sf U},Q_{\sf X|U})-E\}$ is sufficient to approximate the output distribution ${π}_{\sf X}^{\otimes n}$ using the channel $Q_{\sf X|U}^{\otimes n}$, where $Q_{\sf U}\to Q_{\sf X|U}\to Q_{\sf X}$, and is also necessary in the case of finite $\mathcal{U}$ and $\mathcal{X}$. In particular, a randomness rate of $\inf_{Q_{\sf U}}I(Q_{\sf U},Q_{\sf X|U})-E$ is always sufficient. We also study the convergence of the approximation error under the high probability criteria in the case of random codebooks. Moreover, by developing simple bounds relating $E_γ$ and other distance measures, we are able to determine the exact linear growth rate of the approximation errors measured in relative entropy and smooth Rényi divergences for a fixed-input randomness rate. The new resolvability result is then used to derive 1) a one-shot upper bound on the probability of excess distortion in lossy compression, which is exponentially tight in the i.i.d.~setting, 2) a one-shot version of the mutual covering lemma, and 3) a lower bound on the size of the eavesdropper list to include the actual message and a lower bound on the eavesdropper false-alarm probability in the wiretap channel problem, which is (asymptotically) ensemble-tight.
A basic two-terminal secret key generation model is considered, where the interactive communication rate between the terminals may be limited, and in particular may not be enough to achieve the maximum key rate. We first prove a multi-letter characterization of the key-communication rate region (where the number of auxiliary random variables depend on the number of rounds of the communication), and then provide an equivalent but simpler characterization in terms of concave envelopes in the case of unlimited number of rounds. Two extreme cases are given special attention. First, in the regime of very low communication rates, the \emph{key bits per interaction bit} (KBIB) is expressed with a new "symmetric strong data processing constant", which has a concave envelope characterization analogous to that of the conventional strong data processing constant. The symmetric strong data processing constant can be upper bounded by the supremum of the maximal correlation coefficient over a set of distributions, which allows us to determine the KBIB for binary symmetric sources, and conclude, in particular, that the interactive scheme is not more efficient than the one-way scheme at least in the low communication-rate regime. Second, a new characterization of the \emph{minimum interaction rate needed for achieving the maximum key rate} (MIMK) is given, and we resolve a conjecture by Tyagi regarding the MIMK for (possibly nonsymmetric) binary sources. We also propose a new conjecture for binary symmetric sources that the interactive scheme is not more efficient than the one-way scheme at any communication rate.
For each positive integer $n$, let $g_{\mathbb Z}(n)$ be the smallest integer such that if an integral quadratic form in $n$ variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of $g_{\mathbb Z}(n)$ squares of integral linear forms. We show that as $n$ goes to infinity, the growth of $g_{\mathbb Z}(n)$ is at most an exponential of $\sqrt{n}$. Our result improves the best known upper bound on $g_{\mathbb Z}(n)$ which is in the order of an exponential of $n$. We also define an analogous number $g_{\mathcal O}^*(n)$ for writing hermitian forms over the ring of integers $\mathcal O$ of an imaginary quadratic field as sums of norms of integral linear forms, and when the class number of the imaginary quadratic field is 1, we show that the growth of $g_{\mathcal O}^*(n)$ is at most an exponential of $\sqrt{n}$. We also improve results of Conway-Sloane and Kim-Oh on $s$-integral lattices.
A hybrid structure of a graphene nanomesh with the gold nanodisks is studied to enhance the light absorption by localized surface plasmon resonance. From the reflection spectra of the visible range for graphene nanomesh samples without and with nanodisks, it is found that the absorption of graphene nanomesh structures is greatly enhanced in the presence of gold nanodisks around the resonance wavelength. Simulation results based on the finite-difference time-domain method support the experimental observations. This study demonstrates the potential of constructing graphene based photodetectors with a high light absorption efficiency and wavelength selectivity.
We report a highly efficient tunable THz reflector in graphene. By applying a small gate voltage (up to 3 V), the reflectance of graphene is modulated from a minimum of 0.79% to a maximum of 33.4% using graphene/ionic liquid structures at room temperature, and the reflection tuning is uniform within a wide spectral range (0.1 - 1.5 THz). Our observation is explained by the Drude model, which describes the THz wave-induced intraband transition in graphene. This tunable reflectance of graphene may contribute to broadband THz mirrors, deformable THz mirrors, variable THz beam splitters and other optical components.