Universal compression of Gaussian sources with unknown parameters

For a collection of distributions over a countable support set, the worst case universal compression formulation by Shtarkov attempts to assign a universal distribution over the support set. The formulation aims to ensure that the universal distribution does not underestimate the probability of any element in the support set relative to distributions in the collection. When the alphabet is uncountable and we have a collection $\cal P$ of Lebesgue continuous measures instead, we ask if there is a corresponding universal probability density function (pdf) that does not underestimate the value of the density function at any point in the support relative to pdfs in $\cal P$. Analogous to the worst case redundancy of a collection of distributions over a countable alphabet, we define the \textit{attenuation} of a class to be $A$ when the worst case optimal universal pdf at any point $x$ in the support is always at least the value any pdf in the collection $\cal P$ assigns to $x$ divided by $A$. We analyze the attenuation of the worst optimal universal pdf over length-$n$ samples generated \textit{i.i.d.} from a Gaussian distribution whose mean can be anywhere between $-\alpha/2$ to $\alpha/2$ and variance between $\sigma_m^2$ and $\sigma_M^2$. We show that this attenuation is finite, grows with the number of samples as ${\cal O}(n)$, and also specify the attentuation exactly without approximations. When only one parameter is allowed to vary, we show that the attenuation grows as ${\cal O}(\sqrt{n})$, again keeping in line with results from prior literature that fix the order of magnitude as a factor of $\sqrt{n}$ per parameter. In addition, we also specify the attenuation exactly without approximation when only the mean or only the variance is allowed to vary

Inferring macro-ecological patterns from local presence/absence data

Biodiversity provides support for life, vital provisions, regulating services and has positive cultural impacts. It is therefore important to have accurate methods to measure biodiversity, in order to safeguard it when we discover it to be threatened. For practical reasons, biodiversity is usually measured at fine scales whereas diversity issues (e.g. conservation) interest regional or global scales. Moreover, biodiversity may change across spatial scales. It is therefore a key challenge to be able to translate local information on biodiversity into global patterns. Many databases give no information about the abundances of a species within an area, but only its occurrence in each of the surveyed plots. In this paper, we introduce an analytical framework (implemented in a ready‐to‐use R code) to infer species richness and abundances at large spatial scales in biodiversity‐rich ecosystems when species presence/absence information is available on various scattered samples (i.e. upscaling). This framework is based on the scale‐invariance property of the negative binomial. Our approach allows to infer and link within a unique framework important and well‐known biodiversity patterns of ecological theory, such as the species accumulation curve (SAC) and the relative species abundance (RSA) as well as a new emergent pattern, which is the relative species occupancy (RSO). Our estimates are robust and accurate, as confirmed by tests performed on both in silico‐generated and real forests. We demonstrate the accuracy of our predictions using data from two well‐studied forest stands. Moreover, we compared our results with other popular methods proposed in the literature to infer species richness from presence to absence data and we showed that our framework gives better estimates. It has thus important applications to biodiversity research and conservation practice

Towards Quantum Belief Propagation for LDPC Decoding in Wireless Networks

We present Quantum Belief Propagation (QBP), a Quantum Annealing (QA) based decoder design for Low Density Parity Check (LDPC) error control codes, which have found many useful applications in Wi-Fi, satellite communications, mobile cellular systems, and data storage systems. QBP reduces the LDPC decoding to a discrete optimization problem, then embeds that reduced design onto quantum annealing hardware. QBP's embedding design can support LDPC codes of block length up to 420 bits on real state-of-the-art QA hardware with 2,048 qubits. We evaluate performance on real quantum annealer hardware, performing sensitivity analyses on a variety of parameter settings. Our design achieves a bit error rate of $10^{-8}$ in 20 $\mu$s and a 1,500 byte frame error rate of $10^{-6}$ in 50 $\mu$s at SNR 9 dB over a Gaussian noise wireless channel. Further experiments measure performance over real-world wireless channels, requiring 30 $\mu$s to achieve a 1,500 byte 99.99$\%$ frame delivery rate at SNR 15-20 dB. QBP achieves a performance improvement over an FPGA based soft belief propagation LDPC decoder, by reaching a bit error rate of $10^{-8}$ and a frame error rate of $10^{-6}$ at an SNR 2.5--3.5 dB lower. In terms of limitations, QBP currently cannot realize practical protocol-sized ($\textit{e.g.,}$ Wi-Fi, WiMax) LDPC codes on current QA processors. Our further studies in this work present future cost, throughput, and QA hardware trend considerations

On the Communication Complexity of Secure Computation

Information theoretically secure multi-party computation (MPC) is a central primitive of modern cryptography. However, relatively little is known about the communication complexity of this primitive. In this work, we develop powerful information theoretic tools to prove lower bounds on the communication complexity of MPC. We restrict ourselves to a 3-party setting in order to bring out the power of these tools without introducing too many complications. Our techniques include the use of a data processing inequality for residual information - i.e., the gap between mutual information and G\'acs-K\"orner common information, a new information inequality for 3-party protocols, and the idea of distribution switching by which lower bounds computed under certain worst-case scenarios can be shown to apply for the general case. Using these techniques we obtain tight bounds on communication complexity by MPC protocols for various interesting functions. In particular, we show concrete functions that have "communication-ideal" protocols, which achieve the minimum communication simultaneously on all links in the network. Also, we obtain the first explicit example of a function that incurs a higher communication cost than the input length in the secure computation model of Feige, Kilian and Naor (1994), who had shown that such functions exist. We also show that our communication bounds imply tight lower bounds on the amount of randomness required by MPC protocols for many interesting functions.Comment: 37 page

Coded Merkle Tree: Solving Data Availability Attacks in Blockchains

In this paper, we propose coded Merkle tree (CMT), a novel hash accumulator that offers a constant-cost protection against data availability attacks in blockchains, even if the majority of the network nodes are malicious. A CMT is constructed using a family of sparse erasure codes on each layer, and is recovered by iteratively applying a peeling-decoding technique that enables a compact proof for data availability attack on any layer. Our algorithm enables any node to verify the full availability of any data block generated by the system by just downloading a $\Theta(1)$ byte block hash commitment and randomly sampling $\Theta(\log b)$ bytes, where $b$ is the size of the data block. With the help of only one connected honest node in the system, our method also allows any node to verify any tampering of the coded Merkle tree by just downloading $\Theta(\log b)$ bytes. We provide a modular library for CMT in Rust and Python and demonstrate its efficacy inside the Parity Bitcoin client.Comment: To appear in Financial Cryptography and Data Security (FC) 202

Distance Properties of Short LDPC Codes and their Impact on the BP, ML and Near-ML Decoding Performance

Parameters of LDPC codes, such as minimum distance, stopping distance, stopping redundancy, girth of the Tanner graph, and their influence on the frame error rate performance of the BP, ML and near-ML decoding over a BEC and an AWGN channel are studied. Both random and structured LDPC codes are considered. In particular, the BP decoding is applied to the code parity-check matrices with an increasing number of redundant rows, and the convergence of the performance to that of the ML decoding is analyzed. A comparison of the simulated BP, ML, and near-ML performance with the improved theoretical bounds on the error probability based on the exact weight spectrum coefficients and the exact stopping size spectrum coefficients is presented. It is observed that decoding performance very close to the ML decoding performance can be achieved with a relatively small number of redundant rows for some codes, for both the BEC and the AWGN channels