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In this paper, we present a method to encrypt dynamic controllers that can be implemented through most homomorphic encryption schemes, including somewhat, leveled fully, and fully homomorphic encryption. To this end, we represent the output of the given controller as a linear combination of a fixed number of previous inputs and outputs. As a result, the encrypted controller involves only a limited number of homomorphic multiplications on every encrypted data, assuming that the output is re-encrypted and transmitted back from the actuator. A guidance for parameter choice is also provided, ensuring that the encrypted controller achieves predefined performance for an infinite time horizon. Furthermore, we propose a customization of the method for Ring-Learning With Errors (Ring-LWE) based cryptosystems, where a vector of messages can be encrypted into a single ciphertext and operated simultaneously, thus reducing computation and communication loads. Unlike previous results, the proposed customization does not require extra algorithms such as rotation, other than basic addition and multiplication. Simulation results demonstrate the effectiveness of the proposed method.

We propose a cryptography-inspired model for nonlocal correlations. Following the celebrated De Broglie-Bohm theory, we model nonlocal boxes as realistic systems with instantaneous signalling at the hidden variable level. By introducing randomness in the distribution of the hidden variable, the superluminal signalling model is made compatible with the operational no-signalling condition. As the design mimics the famous symmetric key encryption system called {\it One Time Pads} (OTP), we call this the OTP model for nonlocal boxes. We demonstrate utility of this model in several esoteric examples related to the nonclassicality of nonlocal boxes. In particular, the breakdown of communication complexity using nonlocal boxes can be better understood in this framework. Furthermore, we discuss the Van Dam protocol and show its connection to homomorphic encryption in cryptography. We also discuss possible ways of encapsulating quantum realizable nonlocal correlations within this framework and show that the principle of Information Causality imposes further constraints at the hidden variable level. Present work thus orchestrates the results in classical cryptography to improve our understanding of nonlocal correlations and welcomes further research to this connection.

Despite their simple intuition, convolutions are more tedious to analyze than dense layers, which complicates the generalization of theoretical and algorithmic ideas. We provide a new perspective onto convolutions through tensor networks (TNs) which allow reasoning about the underlying tensor multiplications by drawing diagrams, and manipulating them to perform function transformations, sub-tensor access, and fusion. We demonstrate this expressive power by deriving the diagrams of various autodiff operations and popular approximations of second-order information with full hyper-parameter support, batching, channel groups, and generalization to arbitrary convolution dimensions. Further, we provide convolution-specific transformations based on the connectivity pattern which allow to re-wire and simplify diagrams before evaluation. Finally, we probe computational performance, relying on established machinery for efficient TN contraction. Our TN implementation speeds up a recently-proposed KFAC variant up to 4.5x and enables new hardware-efficient tensor dropout for approximate backpropagation.

While many natural language inference (NLI) datasets target certain semantic phenomena, e.g., negation, tense & aspect, monotonicity, and presupposition, to the best of our knowledge, there is no NLI dataset that involves diverse types of spatial expressions and reasoning. We fill this gap by semi-automatically creating an NLI dataset for spatial reasoning, called SpaceNLI. The data samples are automatically generated from a curated set of reasoning patterns, where the patterns are annotated with inference labels by experts. We test several SOTA NLI systems on SpaceNLI to gauge the complexity of the dataset and the system's capacity for spatial reasoning. Moreover, we introduce a Pattern Accuracy and argue that it is a more reliable and stricter measure than the accuracy for evaluating a system's performance on pattern-based generated data samples. Based on the evaluation results we find that the systems obtain moderate results on the spatial NLI problems but lack consistency per inference pattern. The results also reveal that non-projective spatial inferences (especially due to the "between" preposition) are the most challenging ones.

Kolam is a ritual art form practised by people in South India and consists of rule-bound geometric patterns of dots and lines. Single loop Kolams are mathematical closed loop patterns drawn over a grid of dots and conforming to certain heuristics. In this work, we propose a novel encoding scheme where we map the angular movements of Kolam at lattice points into sequences containing $4$ distinct symbols. This is then used to simulate single loop Kolam procedure via turtle moves in accordance with the desired angular direction at specific points. We thus obtain sequential codes for Kolams, unique up to cyclic permutations. We specify the requirements for the algorithm and indicate the general methodology. We demonstrate a sample of Kolams using our algorithm with a software implementation in Python.

The goal of this work is to study waves interacting with partially immersed objects allowed to move freely in the vertical direction, and in a regime in which the propagation of the waves is described by the one dimensional Boussinesq-Abbott system. The problem can be reduced to a transmission problem for this Boussinesq system, in which the transmission conditions between the components of the domain at the left and at the right of the object are determined through the resolution of coupled forced ODEs in time satisfied by the vertical displacement of the object and the average discharge in the portion of the fluid located under the object. We propose a new extended formulation in which these ODEs are complemented by two other forced ODEs satisfied by the trace of the surface elevation at the contact points. The interest of this new extended formulation is that the forcing terms are easy to compute numerically and that the surface elevation at the contact points is furnished for free. Based on this formulation, we propose a second order scheme that involves a generalization of the MacCormack scheme with nonlocal flux and a source term, which is coupled to a second order Heun scheme for the ODEs. In order to validate this scheme, several explicit solutions for this wave-structure interaction problem are derived and can serve as benchmark for future codes. As a byproduct, our method provides a second order scheme for the generation of waves at the entrance of the numerical domain for the Boussinesq-Abbott system.

Over-the-air computation (AirComp), as a data aggregation method that can improve network efficiency by exploiting the superposition characteristics of wireless channels, has received much attention recently. Meanwhile, the orthogonal time frequency space (OTFS) modulation can provide a strong Doppler resilience and facilitates reliable transmission for high-mobility communications. Hence, in this work, we investigate an OTFS-based AirComp system in the presence of time-frequency dual-selective channels. In particular, we commence from the development of a novel transmission framework for the considered system, where the pilot signal is sent together with data and the channel estimation is implemented according to the echo from the access point to the sensor, thereby reducing the overhead of channel state information (CSI) feedback. Hereafter, based on the CSI estimated from the previous frame, a robust precoding matrix aiming at minimizing mean square error in the current frame is designed, which takes into account the estimation error from the receiver noise and the outdated CSI. The simulation results demonstrate the effectiveness of the proposed robust precoding scheme by comparing it with the non-robust precoding. The performance gain is more obvious in high signal-to-noise ratio in case of large channel estimation errors.

Building an accurate surrogate model for the spatio-temporal outputs of a computer simulation is a challenging task. A simple approach to improve the accuracy of the surrogate is to cluster the outputs based on similarity and build a separate surrogate model for each cluster. This clustering is relatively straightforward when the output at each time step is of moderate size. However, when the spatial domain is represented by a large number of grid points, numbering in the millions, the clustering of the data becomes more challenging. In this report, we consider output data from simulations of a jet interacting with high explosives. These data are available on spatial domains of different sizes, at grid points that vary in their spatial coordinates, and in a format that distributes the output across multiple files at each time step of the simulation. We first describe how we bring these data into a consistent format prior to clustering. Borrowing the idea of random projections from data mining, we reduce the dimension of our data by a factor of thousand, making it possible to use the iterative k-means method for clustering. We show how we can use the randomness of both the random projections, and the choice of initial centroids in k-means clustering, to determine the number of clusters in our data set. Our approach makes clustering of extremely high dimensional data tractable, generating meaningful cluster assignments for our problem, despite the approximation introduced in the random projections.

Computer simulations, especially of complex phenomena, can be expensive, requiring high-performance computing resources. Often, to understand a phenomenon, multiple simulations are run, each with a different set of simulation input parameters. These data are then used to create an interpolant, or surrogate, relating the simulation outputs to the corresponding inputs. When the inputs and outputs are scalars, a simple machine learning model can suffice. However, when the simulation outputs are vector valued, available at locations in two or three spatial dimensions, often with a temporal component, creating a surrogate is more challenging. In this report, we use a two-dimensional problem of a jet interacting with high explosives to understand how we can build high-quality surrogates. The characteristics of our data set are unique - the vector-valued outputs from each simulation are available at over two million spatial locations; each simulation is run for a relatively small number of time steps; the size of the computational domain varies with each simulation; and resource constraints limit the number of simulations we can run. We show how we analyze these extremely large data-sets, set the parameters for the algorithms used in the analysis, and use simple ways to improve the accuracy of the spatio-temporal surrogates without substantially increasing the number of simulations required.

The generalization mystery in deep learning is the following: Why do over-parameterized neural networks trained with gradient descent (GD) generalize well on real datasets even though they are capable of fitting random datasets of comparable size? Furthermore, from among all solutions that fit the training data, how does GD find one that generalizes well (when such a well-generalizing solution exists)? We argue that the answer to both questions lies in the interaction of the gradients of different examples during training. Intuitively, if the per-example gradients are well-aligned, that is, if they are coherent, then one may expect GD to be (algorithmically) stable, and hence generalize well. We formalize this argument with an easy to compute and interpretable metric for coherence, and show that the metric takes on very different values on real and random datasets for several common vision networks. The theory also explains a number of other phenomena in deep learning, such as why some examples are reliably learned earlier than others, why early stopping works, and why it is possible to learn from noisy labels. Moreover, since the theory provides a causal explanation of how GD finds a well-generalizing solution when one exists, it motivates a class of simple modifications to GD that attenuate memorization and improve generalization. Generalization in deep learning is an extremely broad phenomenon, and therefore, it requires an equally general explanation. We conclude with a survey of alternative lines of attack on this problem, and argue that the proposed approach is the most viable one on this basis.