Horstein, Burnashev, Shayevitz and Feder, Naghshvar et al. and others have studied sequential transmission of a K-bit message over the binary symmetric channel (BSC) with full, noiseless feedback using posterior matching. Yang et al. provide an improved lower bound on the achievable rate using martingale analysis that relies on the small-enough difference (SED) partitioning introduced by Naghshvar et al. SED requires a relatively complex encoder and decoder. To reduce complexity, this paper replaces SED with relaxed constraints that admit the small enough absolute difference (SEAD) partitioning rule. The main analytical results show that achievable-rate bounds higher than those found by Yang et al. are possible even under the new constraints, which are less restrictive than SED. The new analysis does not use martingale theory for the confirmation phase and applies a surrogate channel technique to tighten the results. An initial systematic transmission further increases the achievable rate bound. The simplified encoder associated with SEAD has a complexity below order O(K^2) and allows simulations for message sizes of at least 1000 bits. For example, simulations achieve 99% of of the channel's 0.50-bit capacity with an average block size of 200 bits for a target codeword error rate of 10^(-3).
相關內容
Regression can be really difficult in case of big datasets, since we have to dealt with huge volumes of data. The demand of computational resources for the modeling process increases as the scale of the datasets does, since traditional approaches for regression involve inverting huge data matrices. The main problem relies on the large data size, and so a standard approach is subsampling that aims at obtaining the most informative portion of the big data. In the current paper we consider an approach based on leverages scores, already existing in the current literature. The aforementioned approach proposed in order to select subdata for linear model discrimination. However, we highlight its importance on the selection of data points that are the most informative for estimating unknown parameters. We conclude that the approach based on leverage scores improves existing approaches, providing simulation experiments as well as a real data application.
We consider several types of internal queries, that is, questions about fragments of a given text $T$ specified in constant space by their locations in $T$. Our main result is an optimal data structure for Internal Pattern Matching (IPM) queries which, given two fragments $x$ and $y$, ask for a representation of all fragments contained in $y$ and matching $x$ exactly; this problem can be viewed as an internal version of the Exact Pattern Matching problem. Our data structure answers IPM queries in time proportional to the quotient $|y|/|x|$ of fragments' lengths, which is required due to the information content of the output. If $T$ is a text of length $n$ over an integer alphabet of size $\sigma$, then our data structure occupies $O(n/ \log_\sigma n)$ machine words (that is, $O(n\log \sigma)$ bits) and admits an $O(n/ \log_\sigma n)$-time construction algorithm. We show the applicability of IPM queries for answering internal queries corresponding to other classic string processing problems. Among others, we derive optimal data structures reporting the periods of a fragment and testing the cyclic equivalence of two fragments. IPM queries have already found numerous further applications, following the path paved by the classic Longest Common Extension (LCE) queries of Landau and Vishkin (JCSS, 1988). In particular, IPM queries have been implemented in grammar-compressed and dynamic settings and, along with LCE queries, constitute elementary operations of the PILLAR model, developed by Charalampopoulos, Kociumaka, and Wellnitz (FOCS 2020). On the way to our main result, we provide a novel construction of string synchronizing sets of Kempa and Kociumaka (STOC 2019). Our method, based on a new restricted version of the recompression technique of Je\.z (J. ACM, 2016), yields a hierarchy of $O(\log n)$ string synchronizing sets covering the whole spectrum of fragments' lengths.
We study the robust communication complexity of maximum matching. Edges of an arbitrary $n$-vertex graph $G$ are randomly partitioned between Alice and Bob independently and uniformly. Alice has to send a single message to Bob such that Bob can find an (approximate) maximum matching of the whole graph $G$. We specifically study the best approximation ratio achievable via protocols where Alice communicates only $\widetilde{O}(n)$ bits to Bob. There has been a growing interest on the robust communication model due to its connections to the random-order streaming model. An algorithm of Assadi and Behnezhad [ICALP'21] implies a $(2/3+\epsilon_0 \sim .667)$-approximation for a small constant $0 < \epsilon_0 < 10^{-18}$, which remains the best-known approximation for general graphs. For bipartite graphs, Assadi and Behnezhad [Random'21] improved the approximation to .716 albeit with a computationally inefficient (i.e., exponential time) protocol. In this paper, we study a natural and efficient protocol implied by a random-order streaming algorithm of Bernstein [ICALP'20] which is based on edge-degree constrained subgraphs (EDCS) [Bernstein and Stein; ICALP'15]. The result of Bernstein immediately implies that this protocol achieves an (almost) $(2/3 \sim .666)$-approximation in the robust communication model. We present a new analysis, proving that it achieves a much better (almost) $(5/6 \sim .833)$-approximation. This significantly improves previous approximations both for general and bipartite graphs. We also prove that our analysis of Bernstein's protocol is tight.
We study the problem of best-arm identification in a distributed variant of the multi-armed bandit setting, with a central learner and multiple agents. Each agent is associated with an arm of the bandit, generating stochastic rewards following an unknown distribution. Further, each agent can communicate the observed rewards with the learner over a bit-constrained channel. We propose a novel quantization scheme called Inflating Confidence for Quantization (ICQ) that can be applied to existing confidence-bound based learning algorithms such as Successive Elimination. We analyze the performance of ICQ applied to Successive Elimination and show that the overall algorithm, named ICQ-SE, has the order-optimal sample complexity as that of the (unquantized) SE algorithm. Moreover, it requires only an exponentially sparse frequency of communication between the learner and the agents, thus requiring considerably fewer bits than existing quantization schemes to successfully identify the best arm. We validate the performance improvement offered by ICQ with other quantization methods through numerical experiments.
In this paper, we find a sample complexity bound for learning a simplex from noisy samples. Assume a dataset of size $n$ is given which includes i.i.d. samples drawn from a uniform distribution over an unknown simplex in $\mathbb{R}^K$, where samples are assumed to be corrupted by a multi-variate additive Gaussian noise of an arbitrary magnitude. We prove the existence of an algorithm that with high probability outputs a simplex having a $\ell_2$ distance of at most $\varepsilon$ from the true simplex (for any $\varepsilon>0$). Also, we theoretically show that in order to achieve this bound, it is sufficient to have $n\ge\left(K^2/\varepsilon^2\right)e^{\Omega\left(K/\mathrm{SNR}^2\right)}$ samples, where $\mathrm{SNR}$ stands for the signal-to-noise ratio. This result solves an important open problem and shows as long as $\mathrm{SNR}\ge\Omega\left(K^{1/2}\right)$, the sample complexity of the noisy regime has the same order to that of the noiseless case. Our proofs are a combination of the so-called sample compression technique in \citep{ashtiani2018nearly}, mathematical tools from high-dimensional geometry, and Fourier analysis. In particular, we have proposed a general Fourier-based technique for recovery of a more general class of distribution families from additive Gaussian noise, which can be further used in a variety of other related problems.
We introduce and analyse a simple probabilistic model of article production and citation behavior that explicitly assumes that there is no decline in citability of a given article over time. It makes predictions about the number and age of items appearing in the reference list of an article. The latter topics have been studied before, but only in the context of data, and to our knowledge no models have been presented. We then perform large-scale analyses of reference list length for a variety of academic disciplines. The results show that our simple model cannot be rejected, and indeed fits the aggregated data on reference lists rather well. Over the last few decades, the relationship between total publications and mean reference list length is linear to a high level of accuracy. Although our model is clearly an oversimplification, it will likely prove useful for further modeling of the scholarly literature. Finally, we connect our work to the large literature on "aging" or "obsolescence" of scholarly publications, and argue that the importance of that area of research is no longer clear, while much of the existing literature is confused and confusing.
Assessing causal effects in the presence of unmeasured confounding is a challenging problem. Although auxiliary variables, such as instrumental variables, are commonly used to identify causal effects, they are often unavailable in practice due to stringent and untestable conditions. To address this issue, previous researches have utilized linear structural equation models to show that the causal effect can be identifiable when noise variables of the treatment and outcome are both non-Gaussian. In this paper, we investigate the problem of identifying the causal effect using auxiliary covariates and non-Gaussianity from the treatment. Our key idea is to characterize the impact of unmeasured confounders using an observed covariate, assuming they are all Gaussian. The auxiliary covariate can be an invalid instrument or an invalid proxy variable. We demonstrate that the causal effect can be identified using this measured covariate, even when the only source of non-Gaussianity comes from the treatment. We then extend the identification results to the multi-treatment setting and provide sufficient conditions for identification. Based on our identification results, we propose a simple and efficient procedure for calculating causal effects and show the $\sqrt{n}$-consistency of the proposed estimator. Finally, we evaluate the performance of our estimator through simulation studies and an application.
Feedback from active galactic nuclei (AGN) and supernovae can affect measurements of integrated SZ flux of halos ($Y_\mathrm{SZ}$) from CMB surveys, and cause its relation with the halo mass ($Y_\mathrm{SZ}-M$) to deviate from the self-similar power-law prediction of the virial theorem. We perform a comprehensive study of such deviations using CAMELS, a suite of hydrodynamic simulations with extensive variations in feedback prescriptions. We use a combination of two machine learning tools (random forest and symbolic regression) to search for analogues of the $Y-M$ relation which are more robust to feedback processes for low masses ($M\lesssim 10^{14}\, h^{-1} \, M_\odot$); we find that simply replacing $Y\rightarrow Y(1+M_*/M_\mathrm{gas})$ in the relation makes it remarkably self-similar. This could serve as a robust multiwavelength mass proxy for low-mass clusters and galaxy groups. Our methodology can also be generally useful to improve the domain of validity of other astrophysical scaling relations. We also forecast that measurements of the $Y-M$ relation could provide percent-level constraints on certain combinations of feedback parameters and/or rule out a major part of the parameter space of supernova and AGN feedback models used in current state-of-the-art hydrodynamic simulations. Our results can be useful for using upcoming SZ surveys (e.g., SO, CMB-S4) and galaxy surveys (e.g., DESI and Rubin) to constrain the nature of baryonic feedback. Finally, we find that the an alternative relation, $Y-M_*$, provides complementary information on feedback than $Y-M$
The well-known discrete Fourier transform (DFT) can easily be generalized to arbitrary nodes in the spatial domain. The fast procedure for this generalization is referred to as nonequispaced fast Fourier transform (NFFT). Various applications such as MRI, solution of PDEs, etc., are interested in the inverse problem, i.e., computing Fourier coefficients from given nonequispaced data. In this paper we survey different kinds of approaches to tackle this problem. In contrast to iterative procedures, where multiple iteration steps are needed for computing a solution, we focus especially on so-called direct inversion methods. We review density compensation techniques and introduce a new scheme that leads to an exact reconstruction for trigonometric polynomials. In addition, we consider a matrix optimization approach using Frobenius norm minimization to obtain an inverse NFFT.
We consider the problem of discovering $K$ related Gaussian directed acyclic graphs (DAGs), where the involved graph structures share a consistent causal order and sparse unions of supports. Under the multi-task learning setting, we propose a $l_1/l_2$-regularized maximum likelihood estimator (MLE) for learning $K$ linear structural equation models. We theoretically show that the joint estimator, by leveraging data across related tasks, can achieve a better sample complexity for recovering the causal order (or topological order) than separate estimations. Moreover, the joint estimator is able to recover non-identifiable DAGs, by estimating them together with some identifiable DAGs. Lastly, our analysis also shows the consistency of union support recovery of the structures. To allow practical implementation, we design a continuous optimization problem whose optimizer is the same as the joint estimator and can be approximated efficiently by an iterative algorithm. We validate the theoretical analysis and the effectiveness of the joint estimator in experiments.
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