We construct six new explicit families of linear maximum sum-rank distance (MSRD) codes, each of which has the smallest field sizes among all known MSRD codes for some parameter regime. Using them and a previous result of the author, we provide two new explicit families of linear partial MDS (PMDS) codes with smaller field sizes than previous PMDS codes for some parameter regimes. Our approach is to characterize evaluation points that turn extended Moore matrices into the parity-check matrix of a linear MSRD code. We then produce such sequences from codes with good Hamming-metric parameters. The six new families of linear MSRD codes with smaller field sizes are obtained using MDS codes, Hamming codes, BCH codes and three Algebraic-Geometry codes. The MSRD codes based on Hamming codes, of minimum sum-rank distance $ 3 $, meet a recent bound by Byrne et al.
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A dynamic mean field theory is developed for model based Bayesian reinforcement learning in the large state space limit. In an analogy with the statistical physics of disordered systems, the transition probabilities are interpreted as couplings, and value functions as deterministic spins, and thus the sampled transition probabilities are considered to be quenched random variables. The results reveal that, under standard assumptions, the posterior over Q-values is asymptotically independent and Gaussian across state-action pairs, for infinite horizon problems. The finite horizon case exhibits the same behaviour for all state-actions pairs at each time but has an additional correlation across time, for each state-action pair. The results also hold for policy evaluation. The Gaussian statistics can be computed from a set of coupled mean field equations derived from the Bellman equation, which we call dynamic mean field programming (DMFP). For Q-value iteration, approximate equations are obtained by appealing to extreme value theory, and closed form expressions are found in the independent and identically distributed case. The Lyapunov stability of these closed form equations is studied.
The main goal of this thesis is to show the crucial role that plays the symbol in analysing the spectrum the sequence of matrices resulting from PDE approximation and in designing a fast method to solve the associated linear problem. In the first part, we study the spectral properties of the matrices arising from $\mathbb{P}_k$ Lagrangian Finite Elements approximation of second order elliptic differential problem with Dirichlet boundary conditions and where the operator is $\mathrm{div} \left(-a(\mathbf{x}) \nabla\cdot\right)$, with $a$ continuous and positive over $\overline \Omega$, $\Omega$ being an open and bounded subset of $\mathbb{R}^d$, $d\ge 1$. We investigate the spectral distribution in the Weyl sense, with a concise overview on localization, clustering, extremal eigenvalues, and asymptotic conditioning. We study in detail the case of constant coefficients on $\Omega=(0,1)^2$ and we give a brief account in the case of variable coefficients and more general domains. While in the second part, we design a fast method of multigrid type for the resolution of linear systems arising from the $\mathbb{Q}_k$ Finite Elements approximation of the same considered problem in one and higher dimensional. The analysis is performed in one dimension, while the numerics are carried out also in higher dimension $d\ge 2$, demonstrating an optimal behavior in terms of the dependency on the matrix size and a robustness with respect to the dimensionality $d$ and to the polynomial degree $k$.
We describe some pseudorandom properties of binary linear codes achieving capacity on the binary erasure channel under bit-MAP decoding (as shown in Kudekar et al this includes doubly transitive codes and, in particular, Reed-Muller codes). We show that for all integer $q \ge 2$ the $\ell_q$ norm of the characteristic function of such 'pseudorandom' code decreases as fast as that of any code of the same rate (and equally fast as that of a random code) under the action of the noise operator. In information-theoretic terms this means that the $q^{th}$ R\'enyi entropy of this code increases as fast as possible over the binary symmetric channel. In particular (taking $q = \infty$) this shows that such codes have the smallest asymptotic undetected error probability (equal to that of a random code) over the BSC, for a certain range of parameters. We also study the number of times a certain local pattern, a 'rhombic' $4$-tuple of codewords, appears in a linear code, and show that for a certain range of parameters this number for pseudorandom codes is similar to that for a random code.
The Erd\H{o}s distinct distance problem is a ubiquitous problem in discrete geometry. Somewhat less well known is Erd\H{o}s' distinct angle problem, the problem of finding the minimum number of distinct angles between $n$ non-collinear points in the plane. Recent work has introduced bounds on a wide array of variants of this problem, inspired by similar variants in the distance setting. In this short note, we improve the best known upper bound for the minimum number of distinct angles formed by $n$ points in general position from $O(n^{\log_2(7)})$ to $O(n^2)$. Before this work, similar bounds relied on projections onto a generic plane from higher dimensional space. In this paper, we employ the geometric properties of a logarithmic spiral, sidestepping the need for a projection.
A sum-rank-metric code attaining the Singleton bound is called maximum sum-rank distance (MSRD). MSRD codes have applications in space-time coding and construction of partial-MDS codes for repair in distributed storage. MSRD codes have been constructed in some parameter cases. In this paper we construct a ${\bf F}_q$-linear MSRD code over some field ${\bf F}_q$ with different matrix sizes $n_1>n_2>\cdots>n_t$ satisfying $n_i \geq n_{i+1}^2+\cdots+n_t^2$ for $i=1, 2, \ldots, t-1$ for any given minimum sum-rank distance. Many good linear sum-rank-metric codes over small fields with such different matrix sizes are given. A lower bound on the dimensions of constructed ${\bf F}_{q^2}$-linear sum-rank-metric codes over ${\bf F}_{q^2}$ with such different matrix sizes and given minimum sum-rank distances is also presented.
This paper studies inference in randomized controlled trials with multiple treatments, where treatment status is determined according to a "matched tuples" design. Here, by a matched tuples design, we mean an experimental design where units are sampled i.i.d. from the population of interest, grouped into "homogeneous" blocks with cardinality equal to the number of treatments, and finally, within each block, each treatment is assigned exactly once uniformly at random. We first study estimation and inference for matched tuples designs in the general setting where the parameter of interest is a vector of linear contrasts over the collection of average potential outcomes for each treatment. Parameters of this form include but are not limited to standard average treatment effects used to compare one treatment relative to another. We first establish conditions under which a sample analogue estimator is asymptotically normal and construct a consistent estimator of its corresponding asymptotic variance. Combining these results establishes the asymptotic validity of tests based on these estimators. In contrast, we show that a common testing procedure based on a linear regression with block fixed effects and the usual heteroskedasticity-robust variance estimator is invalid in the sense that the resulting test may have a limiting rejection probability under the null hypothesis strictly greater than the nominal level. We then apply our results to study the asymptotic properties of what we call "fully-blocked" $2^K$ factorial designs, which are simply matched tuples designs applied to a full factorial experiment. Leveraging our previous results, we establish that our estimator achieves a lower asymptotic variance under the fully-blocked design than that under any stratified factorial design. A simulation study and empirical application illustrate the practical relevance of our results.
This paper investigates the application of physical-layer network coding (PNC) to Industrial Internet-of-Things (IIoT) where a controller and a robot are out of each other's transmission range, and they exchange messages with the assistance of a relay. We particularly focus on a scenario where the controller has more transmitted information, and the channel of the controller is stronger than that of the robot. To reduce the communication latency, we propose an asymmetric transmission scheme where the controller and robot transmit different amount of information in the uplink of PNC simultaneously. To achieve this, the controller chooses a higher order modulation. In addition, the both users apply channel codes to guarantee the reliability. A problem is a superimposed symbol at the relay contains different amount of source information from the two end users. It is thus hard for the relay to deduce meaningful network-coded messages by applying the current PNC decoding techniques which require the end users to transmit the same amount of information. To solve this problem, we propose a lattice-based scheme where the two users encode-and-modulate their information in lattices with different lattice construction levels. Our design is versatile on that the two end users can freely choose their modulation orders based on their channel power, and the design is applicable for arbitrary channel codes.
Experience replay methods, which are an essential part of reinforcement learning(RL) algorithms, are designed to mitigate spurious correlations and biases while learning from temporally dependent data. Roughly speaking, these methods allow us to draw batched data from a large buffer such that these temporal correlations do not hinder the performance of descent algorithms. In this experimental work, we consider the recently developed and theoretically rigorous reverse experience replay (RER), which has been shown to remove such spurious biases in simplified theoretical settings. We combine RER with optimistic experience replay (OER) to obtain RER++, which is stable under neural function approximation. We show via experiments that this has a better performance than techniques like prioritized experience replay (PER) on various tasks, with a significantly smaller computational complexity. It is well known in the RL literature that choosing examples greedily with the largest TD error (as in OER) or forming mini-batches with consecutive data points (as in RER) leads to poor performance. However, our method, which combines these techniques, works very well.
We study the linear contextual bandit problem where an agent has to select one candidate from a pool and each candidate belongs to a sensitive group. In this setting, candidates' rewards may not be directly comparable between groups, for example when the agent is an employer hiring candidates from different ethnic groups and some groups have a lower reward due to discriminatory bias and/or social injustice. We propose a notion of fairness that states that the agent's policy is fair when it selects a candidate with highest relative rank, which measures how good the reward is when compared to candidates from the same group. This is a very strong notion of fairness, since the relative rank is not directly observed by the agent and depends on the underlying reward model and on the distribution of rewards. Thus we study the problem of learning a policy which approximates a fair policy under the condition that the contexts are independent between groups and the distribution of rewards of each group is absolutely continuous. In particular, we design a greedy policy which at each round constructs a ridge regression estimator from the observed context-reward pairs, and then computes an estimate of the relative rank of each candidate using the empirical cumulative distribution function. We prove that the greedy policy achieves, after $T$ rounds, up to log factors and with high probability, a fair pseudo-regret of order $\sqrt{dT}$, where $d$ is the dimension of the context vectors. The policy also satisfies demographic parity at each round when averaged over all possible information available before the selection. We finally show with a proof of concept simulation that our policy achieves sub-linear fair pseudo-regret also in practice.
We revisit the classic regular expression matching problem, that is, given a regular expression $R$ and a string $Q$, decide if $Q$ matches any of the strings specified by $R$. A standard textbook solution [Thompson, CACM 1968] solves this problem in $O(nm)$ time, where $n$ is the length of $Q$ and $m$ is the number of characters in $R$. More recently, several results that improve this bound by polylogarithmic factor have appeared. All of these solutions are essentially based on constructing and simulation a non-deterministic finite automaton. On the other hand, assuming the strong exponential time hypotheses we cannot solve regular expression $O((nm)^{1-\epsilon})$ [Backurs and Indyk, FOCS 2016]. Hence, a natural question is if we can design algorithms that can take advantage of other parameters of the problem to obtain more fine-grained bounds. We present the first algorithm for regular expression matching that can take advantage of sparsity of the automaton simulation. More precisely, we define the \emph{density}, $\Delta$, of the instance to be the total number of states in a simulation of a natural automaton for $R$. The density is always at most $nm+1$ but may be significantly smaller for many typical scenarios, e.g., when a string only matches a small part of the regular expression. Our main result is a new algorithm that solves the problem in $$O\left(\Delta \log \log \frac{nm}{\Delta} + n + m\right)$$ time. This result essentially replaces $nm$ with $\Delta$ in the complexity of regular expression matching. Prior to this work no non-trivial bound in terms of $\Delta$ was known. The key technical contribution is a new linear space representation of the classic position automaton that supports fast state-set transition computation in near-linear time in the size of the input and output state sets.
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