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An improved Singleton-type upper bound is presented for the list decoding radius of linear codes, in terms of the code parameters [n,k,d] and the list size L. L-MDS codes are then defined as codes that attain this bound (under a slightly stronger notion of list decodability), with 1-MDS codes corresponding to ordinary linear MDS codes. Several properties of such codes are presented; in particular, it is shown that the 2-MDS property is preserved under duality. Finally, explicit constructions for 2-MDS codes are presented through generalized Reed-Solomon (GRS) codes.

This letter studies the ergodic mutual information (EMI) of keyhole multiple-input multiple-output (MIMO) channels having finite input signals. At first, the EMI of single-stream transmission is investigated depending on whether the channel state information at the transmitter (CSIT) is available or not. Then, the derived results are extended to the case of multi-stream transmission. For the sake of providing more system insights, asymptotic analyses are performed in the regime of high signal-to-noise ratio (SNR), which suggests that the high-SNR EMI converges to some constant with its rate of convergence (ROC) determined by the diversity order. All the results are validated by numerical simulations and are in excellent agreement.

The mutual information (MI) of Gaussian multi-input multi-output (MIMO) channels has been evaluated by utilizing random matrix theory (RMT) and shown to asymptotically follow Gaussian distribution, where the ergodic mutual information (EMI) converges to a deterministic quantity. However, with non-Gaussian channels, there is a bias between the EMI and its deterministic equivalent (DE), whose evaluation is not available in the literature. This bias of the EMI is related to the bias for the trace of the resolvent in large RMT. In this paper, we first derive the bias for the trace of the resolvent, which is further extended to compute the bias for the linear spectral statistics (LSS). Then, we apply the above results on non-Gaussian MIMO channels to determine the bias for the EMI. It is also proved that the bias for the EMI is $-0.5$ times of that for the variance of the MI. Finally, the derived bias is utilized to modify the central limit theory (CLT) and calculate the outage probability. Numerical results show that the modified CLT significantly outperforms previous methods in approximating the distribution of the MI and improves the accuracy for the outage probability evaluation.

Working constructively, we study continuous directed complete posets (dcpos) and the Scott topology. Our two primary novelties are a notion of intrinsic apartness and a notion of sharp elements. Being apart is a positive formulation of being unequal, similar to how inhabitedness is a positive formulation of nonemptiness. To exemplify sharpness, we note that a lower real is sharp if and only if it is located. Our first main result is that for a large class of continuous dcpos, the Bridges-Vita apartness topology and the Scott topology coincide. Although we cannot expect a tight or cotransitive apartness on nontrivial dcpos, we prove that the intrinsic apartness is both tight and cotransitive when restricted to the sharp elements of a continuous dcpo. These include the strongly maximal elements, as studied by Smyth and Heckmann. We develop the theory of strongly maximal elements highlighting its connection to sharpness and the Lawson topology. Finally, we illustrate the intrinsic apartness, sharpness and strong maximality by considering several natural examples of continuous dcpos: the Cantor and Baire domains, the partial Dedekind reals and the lower reals.

We initiate a systematic study on $\mathit{dynamic}$ $\mathit{influence}$ $\mathit{maximization}$ (DIM). In the DIM problem, one maintains a seed set $S$ of at most $k$ nodes in a dynamically involving social network, with the goal of maximizing the expected influence spread while minimizing the amortized updating cost. We consider two evolution models. In the $\mathit{incremental}$ model, the social network gets enlarged over time and one only introduces new users and establishes new social links, we design an algorithm that achieves $(1-1/e-\epsilon)$-approximation to the optimal solution and has $k \cdot\mathsf{poly}(\log n, \epsilon^{-1})$ amortized running time, which matches the state-of-art offline algorithm with only poly-logarithmic overhead. In the $\mathit{fully}$ $\mathit{dynamic}$ model, users join in and leave, influence propagation gets strengthened or weakened in real time, we prove that under the Strong Exponential Time Hypothesis (SETH), no algorithm can achieve $2^{-(\log n)^{1-o(1)}}$-approximation unless the amortized running time is $n^{1-o(1)}$. On the technical side, we exploit novel adaptive sampling approaches that reduce DIM to the dynamic MAX-k coverage problem, and design an efficient $(1-1/e-\epsilon)$-approximation algorithm for it. Our lower bound leverages the recent developed distributed PCP framework.

We present the quantum algorithm for the Longest Trail Problem. The problem is to search the longest edge-simple path for a graph with $n$ vertexes and $m$ edges. Here edge-simple means no edge occurs in the path twice, but vertexes can occur several times. The running time of our algorithm is $O^*(1.728^m)$.

Counterfactual explanations are usually generated through heuristics that are sensitive to the search's initial conditions. The absence of guarantees of performance and robustness hinders trustworthiness. In this paper, we take a disciplined approach towards counterfactual explanations for tree ensembles. We advocate for a model-based search aiming at "optimal" explanations and propose efficient mixed-integer programming approaches. We show that isolation forests can be modeled within our framework to focus the search on plausible explanations with a low outlier score. We provide comprehensive coverage of additional constraints that model important objectives, heterogeneous data types, structural constraints on the feature space, along with resource and actionability restrictions. Our experimental analyses demonstrate that the proposed search approach requires a computational effort that is orders of magnitude smaller than previous mathematical programming algorithms. It scales up to large data sets and tree ensembles, where it provides, within seconds, systematic explanations grounded on well-defined models solved to optimality.

Despite the success of Generative Adversarial Networks (GANs), mode collapse remains a serious issue during GAN training. To date, little work has focused on understanding and quantifying which modes have been dropped by a model. In this work, we visualize mode collapse at both the distribution level and the instance level. First, we deploy a semantic segmentation network to compare the distribution of segmented objects in the generated images with the target distribution in the training set. Differences in statistics reveal object classes that are omitted by a GAN. Second, given the identified omitted object classes, we visualize the GAN's omissions directly. In particular, we compare specific differences between individual photos and their approximate inversions by a GAN. To this end, we relax the problem of inversion and solve the tractable problem of inverting a GAN layer instead of the entire generator. Finally, we use this framework to analyze several recent GANs trained on multiple datasets and identify their typical failure cases.

With the widespread applications of deep convolutional neural networks (DCNNs), it becomes increasingly important for DCNNs not only to make accurate predictions but also to explain how they make their decisions. In this work, we propose a CHannel-wise disentangled InterPretation (CHIP) model to give the visual interpretation to the predictions of DCNNs. The proposed model distills the class-discriminative importance of channels in networks by utilizing the sparse regularization. Here, we first introduce the network perturbation technique to learn the model. The proposed model is capable to not only distill the global perspective knowledge from networks but also present the class-discriminative visual interpretation for specific predictions of networks. It is noteworthy that the proposed model is able to interpret different layers of networks without re-training. By combining the distilled interpretation knowledge in different layers, we further propose the Refined CHIP visual interpretation that is both high-resolution and class-discriminative. Experimental results on the standard dataset demonstrate that the proposed model provides promising visual interpretation for the predictions of networks in image classification task compared with existing visual interpretation methods. Besides, the proposed method outperforms related approaches in the application of ILSVRC 2015 weakly-supervised localization task.

While Generative Adversarial Networks (GANs) have empirically produced impressive results on learning complex real-world distributions, recent work has shown that they suffer from lack of diversity or mode collapse. The theoretical work of Arora et al.~\cite{AroraGeLiMaZh17} suggests a dilemma about GANs' statistical properties: powerful discriminators cause overfitting, whereas weak discriminators cannot detect mode collapse. In contrast, we show in this paper that GANs can in principle learn distributions in Wasserstein distance (or KL-divergence in many cases) with polynomial sample complexity, if the discriminator class has strong distinguishing power against the particular generator class (instead of against all possible generators). For various generator classes such as mixture of Gaussians, exponential families, and invertible neural networks generators, we design corresponding discriminators (which are often neural nets of specific architectures) such that the Integral Probability Metric (IPM) induced by the discriminators can provably approximate the Wasserstein distance and/or KL-divergence. This implies that if the training is successful, then the learned distribution is close to the true distribution in Wasserstein distance or KL divergence, and thus cannot drop modes. Our preliminary experiments show that on synthetic datasets the test IPM is well correlated with KL divergence, indicating that the lack of diversity may be caused by the sub-optimality in optimization instead of statistical inefficiency.