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We study the emptiness and $\lambda$-reachability problems for unary and binary Probabilistic Finite Automata (PFA) and characterise the complexity of these problems in terms of the degree of ambiguity of the automaton and the size of its alphabet. Our main result is that emptiness and $\lambda$-reachability are solvable in EXPTIME for polynomially ambiguous unary PFA and if, in addition, the transition matrix is binary, we show they are in NP. In contrast to the Skolem-hardness of the $\lambda$-reachability and emptiness problems for exponentially ambiguous unary PFA, we show that these problems are NP-hard even for finitely ambiguous unary PFA. For binary polynomially ambiguous PFA with fixed and commuting transition matrices, we prove NP-hardness of the $\lambda$-reachability (dimension 9), nonstrict emptiness (dimension 37) and strict emptiness (dimension 40) problems.

We propose polar encoding, a representation of categorical and numerical $[0,1]$-valued attributes with missing values to be used in a classification context. We argue that this is a good baseline approach, because it can be used with any classification algorithm, preserves missingness information, is very simple to apply and offers good performance. In particular, unlike the existing missing-indicator approach, it does not require imputation, ensures that missing values are equidistant from non-missing values, and lets decision tree algorithms choose how to split missing values, thereby providing a practical realisation of the "missingness incorporated in attributes" (MIA) proposal. Furthermore, we show that categorical and $[0,1]$-valued attributes can be viewed as special cases of a single attribute type, corresponding to the classical concept of barycentric coordinates, and that this offers a natural interpretation of polar encoding as a fuzzified form of one-hot encoding. With an experiment based on twenty real-life datasets with missing values, we show that, in terms of the resulting classification performance, polar encoding performs better than the state-of-the-art strategies \e{multiple imputation by chained equations} (MICE) and \e{multiple imputation with denoising autoencoders} (MIDAS) and -- depending on the classifier -- about as well or better than mean/mode imputation with missing-indicators.

We show that any bounded integral function $f : A \times B \mapsto \{0,1, \dots, \Delta\}$ with rank $r$ has deterministic communication complexity $\Delta^{O(\Delta)} \cdot \sqrt{r} \cdot \log^2 r$, where the rank of $f$ is defined to be the rank of the $A \times B$ matrix whose entries are the function values. As a corollary, we show that any $n$-dimensional polytope that admits a slack matrix with entries from $\{0,1,\dots,\Delta\}$ has extension complexity at most $\exp(\Delta^{O(\Delta)} \cdot \sqrt{n} \cdot \log^2 n)$.

Predictive algorithms are often trained by optimizing some loss function, to which regularization functions are added to impose a penalty for violating constraints. As expected, the addition of such regularization functions can change the minimizer of the objective. It is not well-understood which regularizers change the minimizer of the loss, and, when the minimizer does change, how it changes. We use property elicitation to take first steps towards understanding the joint relationship between the loss and regularization functions and the optimal decision for a given problem instance. In particular, we give a necessary and sufficient condition on loss and regularizer pairs for when a property changes with the addition of the regularizer, and examine some regularizers satisfying this condition standard in the fair machine learning literature. We empirically demonstrate how algorithmic decision-making changes as a function of both data distribution changes and hardness of the constraints.

In this paper, we considier the limiting distribution of the maximum interpoint Euclidean distance $M_n=\max _{1 \leq i<j \leq n}\left\|\boldsymbol{X}_i-\boldsymbol{X}_j\right\|$, where $\boldsymbol{X}_1, \boldsymbol{X}_2, \ldots, \boldsymbol{X}_n$ be a random sample coming from a $p$-dimensional population with dependent sub-gaussian components. When the dimension tends to infinity with the sample size, we proves that $M_n^2$ under a suitable normalization asymptotically obeys a Gumbel type distribution. The proofs mainly depend on the Stein-Chen Poisson approximation method and high dimensional Gaussian approximation.

Existing out-of-distribution (OOD) methods have shown great success on balanced datasets but become ineffective in long-tailed recognition (LTR) scenarios where 1) OOD samples are often wrongly classified into head classes and/or 2) tail-class samples are treated as OOD samples. To address these issues, current studies fit a prior distribution of auxiliary/pseudo OOD data to the long-tailed in-distribution (ID) data. However, it is difficult to obtain such an accurate prior distribution given the unknowingness of real OOD samples and heavy class imbalance in LTR. A straightforward solution to avoid the requirement of this prior is to learn an outlier class to encapsulate the OOD samples. The main challenge is then to tackle the aforementioned confusion between OOD samples and head/tail-class samples when learning the outlier class. To this end, we introduce a novel calibrated outlier class learning (COCL) approach, in which 1) a debiased large margin learning method is introduced in the outlier class learning to distinguish OOD samples from both head and tail classes in the representation space and 2) an outlier-class-aware logit calibration method is defined to enhance the long-tailed classification confidence. Extensive empirical results on three popular benchmarks CIFAR10-LT, CIFAR100-LT, and ImageNet-LT demonstrate that COCL substantially outperforms state-of-the-art OOD detection methods in LTR while being able to improve the classification accuracy on ID data. Code is available at //github.com/mala-lab/COCL.

Debugging physical computing projects provides a rich context to understand cross-disciplinary problem solving that integrates multiple domains of computing and engineering. Yet understanding and assessing students' learning of debugging remains a challenge, particularly in understudied areas such as physical computing, since finding and fixing hardware and software bugs is a deeply contextual practice. In this paper we draw on the rich history of clinical interviews to develop and pilot "failure artifact scenarios" in order to study changes in students' approaches to debugging and troubleshooting electronic textiles (e-textiles). We applied this clinical interview protocol before and after an eight-week-long e-textiles unit. We analyzed pre/post clinical interviews from 18 students at four different schools. The analysis revealed that students improved in identifying bugs with greater specificity, and across domains, and in considering multiple causes for bugs. We discuss implications for developing tools to assess students' debugging abilities through contextualized debugging scenarios in physical computing.

We consider relational semantics (R-models) for the Lambek calculus extended with intersection and explicit constants for zero and unit. For its variant without constants and a restriction which disallows empty antecedents, Andreka and Mikulas (1994) prove strong completeness. We show that it fails without this restriction, but, on the other hand, prove weak completeness for non-standard interpretation of constants. For the standard interpretation, even weak completeness fails. The weak completeness result extends to an infinitary setting, for so-called iterative divisions (Kleene star under division). We also prove strong completeness results for product-free fragments.

These are self-contained lecture notes for spectral independence. For an $n$-vertex graph, the spectral independence condition is a bound on the maximum eigenvalue of the $n\times n$ influence matrix whose entries capture the influence between pairs of vertices, it is closely related to the covariance matrix. We will present recent results showing that spectral independence implies the mixing time of the Glauber dynamics is polynomial (where the degree of the polynomial depends on certain parameters). The proof utilizes local-to-global theorems which we will detail in these notes. Finally, we will present more recent results showing that spectral independence implies an optimal bound on the relaxation time (inverse spectral gap) and with some additional conditions implies an optimal mixing time bound of $O(n\log{n})$ for the Glauber dynamics. We also present the results of Anari, Liu, Oveis Gharan, and Vinzant (2019) for generating a random basis of a matroid. The analysis of the associated bases-exchange walk utilizes the local-to-global theorems used for spectral independence with the Trickle-Down Theorem of Oppenheim (2018) to analyze the local walks. Our focus in these notes is on the analysis of the spectral gap of the associated Markov chains from a functional analysis perspective, and we present proofs of the associated local-to-global theorems from this same Markov chain perspective.

We introduce a generic framework that reduces the computational cost of object detection while retaining accuracy for scenarios where objects with varied sizes appear in high resolution images. Detection progresses in a coarse-to-fine manner, first on a down-sampled version of the image and then on a sequence of higher resolution regions identified as likely to improve the detection accuracy. Built upon reinforcement learning, our approach consists of a model (R-net) that uses coarse detection results to predict the potential accuracy gain for analyzing a region at a higher resolution and another model (Q-net) that sequentially selects regions to zoom in. Experiments on the Caltech Pedestrians dataset show that our approach reduces the number of processed pixels by over 50% without a drop in detection accuracy. The merits of our approach become more significant on a high resolution test set collected from YFCC100M dataset, where our approach maintains high detection performance while reducing the number of processed pixels by about 70% and the detection time by over 50%.