Recently, formal verification of deep neural networks (DNNs) has garnered considerable attention, and over-approximation based methods have become popular due to their effectiveness and efficiency. However, these strategies face challenges in addressing the "unknown dilemma" concerning whether the exact output region or the introduced approximation error violates the property in question. To address this, this paper introduces the UR4NNV verification framework, which utilizes under-approximation reachability analysis for DNN verification for the first time. UR4NNV focuses on DNNs with Rectified Linear Unit (ReLU) activations and employs a binary tree branch-based under-approximation algorithm. In each epoch, UR4NNV under-approximates a sub-polytope of the reachable set and verifies this polytope against the given property. Through a trial-and-error approach, UR4NNV effectively falsifies DNN properties while providing confidence levels when reaching verification epoch bounds and failing falsifying properties. Experimental comparisons with existing verification methods demonstrate the effectiveness and efficiency of UR4NNV, significantly reducing the impact of the "unknown dilemma".

Some Developments in Clustering Analysis on Stochastic Processes Qidi Peng Nan Rao † Ran Zhao ‡ Abstract We review some developments on clustering stochastic processes and come with the conclusion that asymptotically consistent clustering algorithms can be obtained when the processes are ergodic and the dissimilarity measure satisfies the triangle inequality. Examples are provided when the processes are distribution ergodic, covariance ergodic and locally asymptotically self-similar, respectively. Keywords: stochastic process, unsupervised clustering, stationary ergodic processes, local asymptotic self-similarity 1 Introduction A stochastic process is an infinite sequence of random variables indexed by "time". The time indexes can be either discrete or continuous. Stochastic process type data have been broadly explored in biological and medical research (Damian et al., 2007; Zhao et al., 2014; J a askinen et al., 2014; et al., 2018).

Technical and fundamental analysis are traditional tools used to analyze individual stocks; however, the finance literature has shown that the price movement of each individual stock correlates heavily with other stocks, especially those within the same sector. In this paper we propose a general-purpose market representation that incorporates fundamental and technical indicators and relationships between individual stocks. We treat the daily stock market as a ‘market image’ where rows (grouped by market sector) represent individual stocks and columns represent indicators. We apply a convolutional neural network over this market image to build market features in a hierarchical way. We use a recurrent neural network, with an attention mechanism over the market feature maps, to model temporal dynamics in the market. We show that our proposed model outperforms strong baselines in both short-term and long-term stock return prediction tasks. We also show another use for our market image: to construct concise and dense market embeddings suitable for downstream prediction tasks.

In this paper, we design algorithms for clustering locally asymptotically self-similar stochastic processes. We show a sufficient condition on the dissimilarity measure that leads to the consistency of the algorithms for clustering offline and online data settings, respectively. As an example of application, clustering synthetic data sampled from multifractional Brownian motions is provided.

We introduce a new unsupervised learning problem: clustering wide-sense stationary ergodic stochastic processes. A covariance-based dissimilarity measure and consistent algorithms are designed for clustering offline and online data settings, respectively. We also suggest a formal criterion on the efficiency of dissimilarity measures, and discuss of some approach to improve the efficiency of clustering algorithms, when they are applied to cluster particular type of processes, such as self-similar processes with wide-sense stationary ergodic increments. Clustering synthetic data sampled from fractional Brownian motions is provided as an example of application.

While recent neural encoder-decoder models have shown great promise in modeling open-domain conversations, they often generate dull and generic responses. Unlike past work that has focused on diversifying the output of the decoder at word-level to alleviate this problem, we present a novel framework based on conditional variational autoencoders that captures the discourse-level diversity in the encoder. Our model uses latent variables to learn a distribution over potential conversational intents and generates diverse responses using only greedy decoders. We have further developed a novel variant that is integrated with linguistic prior knowledge for better performance. Finally, the training procedure is improved by introducing a bag-of-word loss. Our proposed models have been validated to generate significantly more diverse responses than baseline approaches and exhibit competence in discourse-level decision-making.