In some rate-distortion-type problems, the required fidelity of information is affected by past actions. As a result, the distortion function depends not only on the instantaneous distortion between a source symbol and its representation symbol, but also on past representations. In this paper, we give a formal definition of this problem and introduce both inner (achievable) and outer bounds on the rate-distortion tradeoff. We also discuss convexification of the problem, which makes it easier to find bounds. Problems of this type arise in biological information processing, as well as in recommendation engines; we provide an example applied to a simplified biological information processing problem.

We introduce a two-parameter family of discrepancy measures, termed \emph{$(G,f)$-divergences}, obtained by applying a non-decreasing function $G$ to an $f$-divergence $D_f$. Building on Csiszár's formulation of mutual $f$-information, we define a corresponding $(G,f)$-information measure $ I_{G,f}(X;Y)$. A central theme of the paper is subadditivity over product distributions and product channels. We develop reduction principles showing that, for broad classes of $G$, it suffices to verify divergence subadditivity on binary alphabets. Specializing to the functions $G(x)\in\{x,\log(1+x),-\log(1-x)\}$, we derive tractable sufficient conditions on $f$ that guarantee subadditivity, covering many standard $f$-divergences. Finally, we present applications to finite-blocklength converses for channel coding, bounds in binary hypothesis testing, and an extension of the Shannon--Gallager--Berlekamp sphere-packing exponent framework to subadditive $(G,f)$-divergences.

We consider the source model key agreement problem involving two legitimate parties and an eavesdropper who observe n i.i.d. samples of X, Y, and Z, respectively. In this paper, we focus on one of the simplest instances where the key capacity remains open, specifically when X and Y are binary random variables and Z is a function of the pair (X, Y). The best-known upper bound on the key capacity is characterized by an inf-max optimization problem that generally lacks a closed-form solution. We provide general conditions under which the upper bound reduces to I(X;Y). As an example, we consider the XOR setting in which X and Y are binary, and Z is the XOR of X and Y. The upper bound reduces to I(X;Y) for this source. Next, we conjecture that the rate I(X;Y) is not achievable for the XOR source and provide some ideas that might be useful for developing a new upper bound on the source model problem.

We consider the source model key agreement problem involving two legitimate parties and an eavesdropper who observe n i.i.d. samples of X and Y and Z respectively. In this paper, we focus on one of the simplest instances where the key capacity remains open, specifically when X and Y are binary random variables and Z is a function of the pair ($X, Y$). The best-known upper bound on the key capacity is characterized by an inf-max optimization problem that generally lacks a closedform solution. In this paper, we solve the optimization for some class of sources, thereby providing simple expressions for the upper bound. We provide general conditions under which the upper bound reduces to$I(X; Y)$. As an example, we consider the XOR setting in which X and Y are binary, and Z is the XOR of X and Y. The upper bound reduces to$I(X; Y)$for this source. Next, we conjecture that the rate$I(X; Y)$is not achievable for the XOR source, and provide some ideas that might be useful for developing a new upper bound.${ }^{1}$

A key element in the performance of a compressed sensing (CS) setup is the so called sensing matrix. It is known that the success of CS-based methods strongly rely on the properties of the employed sensing matrix. Random matrices are the widely-adopted choice due to their order-optimal performance and flexibility in size. In real world applications , however, random structures are rarely feasible. For this reason, the deterministic design of sensing matrices has been an ongoing research topic. In this paper, we introduce new classes of deterministic complex-valued sensing matrices based on algebraic curves. In particular, we design a number of algebraic-geometric codes with large minimum distances specifically for the construction of sensing matrices. Our approach is to find maximal curves in the Galois field F p m ‾ , and transform them into F p codes by a trace map. Invoking the Riemann-Roch theorem, we demonstrate that the resulting code has a large minimum distance compared to its length, which leads to a sensing matrix with small coherence value. For general m × n matrices, the Welch bound ( ≈ 1 m ) sets a universal lower-bound on the coherence value, and the bound is achievable only when n ≤ m 2 . In our designs, we are able to construct m × n matrices with n ranging from around 8 m to values larger than O ( m 2 ) by tuning the parameters. Meanwhile, the coherence of the designed matrices differ from the Welch bound by only an O ( log m ) factor. Simulation results indicate that the performance of our matrices in recovering sparse vectors from compressed measurements is superior or equivalent to Gaussian random matrices.

One of the main challenges in diffusion-based molecular communication is dealing with the non-linearity of reaction-diffusion chemical equations. While numerical methods can be used to solve these equations, a change in the input signals or the parameters of the medium requires one to redo the simulations. This makes it difficult to design modulation schemes and practically impossible to prove the optimality of a given transmission strategy. In this paper, we provide an analytical technique for modeling the non-linearity of chemical reaction equations based on the perturbation method. The perturbation method expresses the solution in terms of an infinite power series. An approximate solution can be found by keeping the leading terms of the power series. The approximate solution is shown to track the true solution if either the simulation time interval or the reaction rate is sufficiently small. Approximate solutions for long time intervals are also discussed. An illustrative example is given. For this example, it is shown that when the reaction rate (or the total time interval) is low, instead of using a continuous release waveform, it is optimal for the transmitters to release molecules at two time instances.

In this paper, we propose a new method to reconstruct a signal corrupted by noise where both signal and noise are sparse but in different domains. The main contribution of our algorithm is its low complexity; it has much lower run-time than most other algorithms. The reconstruction quality of our algorithm is both objectively (in terms of PSNR and SSIM) and subjectively better or comparable to other state-of-the-art algorithms. We provide a cost function for our problem, present an iterative method to find its local minimum, and provide the analysis of the algorithm. As an application of this problem, we apply our algorithm for Salt-and-Pepper noise (SPN) and Random-Valued Impulsive Noise (RVIN) removal from images and compare our results with other notable algorithms in the literature. Furthermore, we apply our algorithm for removing clicks from audio signals. Simulation results show that our algorithms are simple and fast, and it outperforms other state-of-the-art methods in terms of reconstruction quality and/or complexity.

One of the main challenges in diffusion-based molecular communication is dealing with the non-linearity of reaction diffusion chemical equations. While numerical methods can be used to solve these equations, a change in the input signals or in the parameters of the medium requires one to redo the simulations. This makes it difficult to design modulation schemes and practically impossible to prove the optimality of a given transmission strategy. We provide an analytical technique for modeling non-linearity of chemical reaction equations based on the perturbation method. It is shown that at low reaction rates, instead of using a continuous transmission waveform, it is optimal for each transmitter to release molecules in a sequence of spikes.