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We revisit constructions based on triads of conics with foci at pairs of vertices of a reference triangle. We find that their 6 vertices lie on well-known conics, whose type we analyze. We give conditions for these to be circles and/or degenerate. In the latter case, we study the locus of their center.

We consider the task of lexicographic direct access to query answers. That is, we want to simulate an array containing the answers of a join query sorted in a lexicographic order chosen by the user. A recent dichotomy showed for which queries and orders this task can be done in polylogarithmic access time after quasilinear preprocessing, but this dichotomy does not tell us how much time is required in the cases classified as hard. We determine the preprocessing time needed to achieve polylogarithmic access time for all self-join free queries and all lexicographical orders. To this end, we propose a decomposition-based general algorithm for direct access on join queries. We then explore its optimality by proving lower bounds for the preprocessing time based on the hardness of a certain online Set-Disjointness problem, which shows that our algorithm's bounds are tight for all lexicographic orders on self-join free queries. Then, we prove the hardness of Set-Disjointness based on the Zero-Clique Conjecture which is an established conjecture from fine-grained complexity theory. We also show that similar techniques can be used to prove that, for enumerating answers to Loomis-Whitney joins, it is not possible to significantly improve upon trivially computing all answers at preprocessing. This, in turn, gives further evidence (based on the Zero-Clique Conjecture) to the enumeration hardness of self-join free cyclic joins with respect to linear preprocessing and constant delay.

We derive conditions for the existence of fixed points of nonnegative neural networks, an important research objective to understand the behavior of neural networks in modern applications involving autoencoders and loop unrolling techniques, among others. In particular, we show that neural networks with nonnegative inputs and nonnegative parameters can be recognized as monotonic and (weakly) scalable functions within the framework of nonlinear Perron-Frobenius theory. This fact enables us to derive conditions for the existence of a nonempty fixed point set of the nonnegative neural networks, and these conditions are weaker than those obtained recently using arguments in convex analysis, which are typically based on the assumption of nonexpansivity of the activation functions. Furthermore, we prove that the shape of the fixed point set of monotonic and weakly scalable neural networks is often an interval, which degenerates to a point for the case of scalable networks. The chief results of this paper are verified in numerical simulations, where we consider an autoencoder-type network that first compresses angular power spectra in massive MIMO systems, and, second, reconstruct the input spectra from the compressed signals.

This paper presents local minimax regret lower bounds for adaptively controlling linear-quadratic-Gaussian (LQG) systems. We consider smoothly parametrized instances and provide an understanding of when logarithmic regret is impossible which is both instance specific and flexible enough to take problem structure into account. This understanding relies on two key notions: That of local-uninformativeness; when the optimal policy does not provide sufficient excitation for identification of the optimal policy, and yields a degenerate Fisher information matrix; and that of information-regret-boundedness, when the small eigenvalues of a policy-dependent information matrix are boundable in terms of the regret of that policy. Combined with a reduction to Bayesian estimation and application of Van Trees' inequality, these two conditions are sufficient for proving regret bounds on order of magnitude $\sqrt{T}$ in the time horizon, $T$. This method yields lower bounds that exhibit tight dimensional dependencies and scale naturally with control-theoretic problem constants. For instance, we are able to prove that systems operating near marginal stability are fundamentally hard to learn to control. We further show that large classes of systems satisfy these conditions, among them any state-feedback system with both $A$- and $B$-matrices unknown. Most importantly, we also establish that a nontrivial class of partially observable systems, essentially those that are over-actuated, satisfy these conditions, thus providing a $\sqrt{T}$ lower bound also valid for partially observable systems. Finally, we turn to two simple examples which demonstrate that our lower bound captures classical control-theoretic intuition: our lower bounds diverge for systems operating near marginal stability or with large filter gain -- these can be arbitrarily hard to (learn to) control.

Stochastic majorization-minimization (SMM) is an online extension of the classical principle of majorization-minimization, which consists of sampling i.i.d. data points from a fixed data distribution and minimizing a recursively defined majorizing surrogate of an objective function. In this paper, we introduce stochastic block majorization-minimization, where the surrogates can now be only block multi-convex and a single block is optimized at a time within a diminishing radius. Relaxing the standard strong convexity requirements for surrogates in SMM, our framework gives wider applicability including online CANDECOMP/PARAFAC (CP) dictionary learning and yields greater computational efficiency especially when the problem dimension is large. We provide an extensive convergence analysis on the proposed algorithm, which we derive under possibly dependent data streams, relaxing the standard i.i.d. assumption on data samples. We show that the proposed algorithm converges almost surely to the set of stationary points of a nonconvex objective under constraints at a rate $O((\log n)^{1+\eps}/n^{1/2})$ for the empirical loss function and $O((\log n)^{1+\eps}/n^{1/4})$ for the expected loss function, where $n$ denotes the number of data samples processed. Under some additional assumption, the latter convergence rate can be improved to $O((\log n)^{1+\eps}/n^{1/2})$. Our results provide first convergence rate bounds for various online matrix and tensor decomposition algorithms under a general Markovian data setting.

The manufacturing industry is currently witnessing a paradigm shift with the unprecedented adoption of industrial robots, and machine vision is a key perception technology that enables these robots to perform precise operations in unstructured environments. However, the sensitivity of conventional vision sensors to lighting conditions and high-speed motion sets a limitation on the reliability and work-rate of production lines. Neuromorphic vision is a recent technology with the potential to address the challenges of conventional vision with its high temporal resolution, low latency, and wide dynamic range. In this paper and for the first time, we propose a novel neuromorphic vision based controller for faster and more reliable machining operations, and present a complete robotic system capable of performing drilling tasks with sub-millimeter accuracy. Our proposed system localizes the target workpiece in 3D using two perception stages that we developed specifically for the asynchronous output of neuromorphic cameras. The first stage performs multi-view reconstruction for an initial estimate of the workpiece's pose, and the second stage refines this estimate for a local region of the workpiece using circular hole detection. The robot then precisely positions the drilling end-effector and drills the target holes on the workpiece using a combined position-based and image-based visual servoing approach. The proposed solution is validated experimentally for drilling nutplate holes on workpieces placed arbitrarily in an unstructured environment with uncontrolled lighting. Experimental results prove the effectiveness of our solution with an average positional errors of less than 0.1 mm, and demonstrate that the use of neuromorphic vision overcomes the lighting and speed limitations of conventional cameras.

Motivated by the goal of achieving long-term drift-free camera pose estimation in complex scenarios, we propose a global positioning framework fusing visual, inertial and Global Navigation Satellite System (GNSS) measurements in multiple layers. Different from previous loosely- and tightly- coupled methods, the proposed multi-layer fusion allows us to delicately correct the drift of visual odometry and keep reliable positioning while GNSS degrades. In particular, local motion estimation is conducted in the inner-layer, solving the problem of scale drift and inaccurate bias estimation in visual odometry by fusing the velocity of GNSS, pre-integration of Inertial Measurement Unit (IMU) and camera measurement in a tightly-coupled way. The global localization is achieved in the outer-layer, where the local motion is further fused with GNSS position and course in a long-term period in a loosely-coupled way. Furthermore, a dedicated initialization method is proposed to guarantee fast and accurate estimation for all state variables and parameters. We give exhaustive tests of the proposed framework on indoor and outdoor public datasets. The mean localization error is reduced up to 63%, with a promotion of 69% in initialization accuracy compared with state-of-the-art works. We have applied the algorithm to Augmented Reality (AR) navigation, crowd sourcing high-precision map update and other large-scale applications.

In this paper we discuss a reduced basis method for linear evolution PDEs, which is based on the application of the Laplace transform. The main advantage of this approach consists in the fact that, differently from time stepping methods, like Runge-Kutta integrators, the Laplace transform allows to compute the solution directly at a given instant, which can be done by approximating the contour integral associated to the inverse Laplace transform by a suitable quadrature formula. In terms of the reduced basis methodology, this determines a significant improvement in the reduction phase - like the one based on the classical proper orthogonal decomposition (POD) - since the number of vectors to which the decomposition applies is drastically reduced as it does not contain all intermediate solutions generated along an integration grid by a time stepping method. We show the effectiveness of the method by some illustrative parabolic PDEs arising from finance and also provide some evidence that the method we propose, when applied to a simple advection equation, does not suffer the problem of slow decay of singular values which instead affects methods based on time integration of the Cauchy problem arising from space discretization.

Many-user MAC is an important model for understanding energy efficiency of massive random access in 5G and beyond. Introduced in Polyanskiy'2017 for the AWGN channel, subsequent works have provided improved bounds on the asymptotic minimum energy-per-bit required to achieve a target per-user error at a given user density and payload, going beyond the AWGN setting. The best known rigorous bounds use spatially coupled codes along with the optimal AMP algorithm. But these bounds are infeasible to compute beyond a few (around 10) bits of payload. In this paper, we provide new achievability bounds for the many-user AWGN and quasi-static Rayleigh fading MACs using the spatially coupled codebook design along with a scalar AMP algorithm. The obtained bounds are computable even up to 100 bits and outperform the previous ones at this payload.

We study the problem of learning in the stochastic shortest path (SSP) setting, where an agent seeks to minimize the expected cost accumulated before reaching a goal state. We design a novel model-based algorithm EB-SSP that carefully skews the empirical transitions and perturbs the empirical costs with an exploration bonus to guarantee both optimism and convergence of the associated value iteration scheme. We prove that EB-SSP achieves the minimax regret rate $\widetilde{O}(B_{\star} \sqrt{S A K})$, where $K$ is the number of episodes, $S$ is the number of states, $A$ is the number of actions and $B_{\star}$ bounds the expected cumulative cost of the optimal policy from any state, thus closing the gap with the lower bound. Interestingly, EB-SSP obtains this result while being parameter-free, i.e., it does not require any prior knowledge of $B_{\star}$, nor of $T_{\star}$ which bounds the expected time-to-goal of the optimal policy from any state. Furthermore, we illustrate various cases (e.g., positive costs, or general costs when an order-accurate estimate of $T_{\star}$ is available) where the regret only contains a logarithmic dependence on $T_{\star}$, thus yielding the first horizon-free regret bound beyond the finite-horizon MDP setting.