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We characterize the geometry and topology of the set of all weight vectors for which a linear neural network computes the same linear transformation $W$. This set of weight vectors is called the fiber of $W$ (under the matrix multiplication map), and it is embedded in the Euclidean weight space of all possible weight vectors. The fiber is an algebraic variety that is not necessarily a manifold. We describe a natural way to stratify the fiber--that is, to partition the algebraic variety into a finite set of manifolds of varying dimensions called strata. We call this set of strata the rank stratification. We derive the dimensions of these strata and the relationships by which they adjoin each other. Although the strata are disjoint, their closures are not. Our strata satisfy the frontier condition: if a stratum intersects the closure of another stratum, then the former stratum is a subset of the closure of the latter stratum. Each stratum is a manifold of class $C^\infty$ embedded in weight space, so it has a well-defined tangent space and normal space at every point (weight vector). We show how to determine the subspaces tangent to and normal to a specified stratum at a specified point on the stratum, and we construct elegant bases for those subspaces. To help achieve these goals, we first derive what we call a Fundamental Theorem of Linear Neural Networks, analogous to what Strang calls the Fundamental Theorem of Linear Algebra. We show how to decompose each layer of a linear neural network into a set of subspaces that show how information flows through the neural network. Each stratum of the fiber represents a different pattern by which information flows (or fails to flow) through the neural network. The topology of a stratum depends solely on this decomposition. So does its geometry, up to a linear transformation in weight space., Comment: 99 pages, 14 figures

Ultra-wideband (UWB) is emerging as a promising solution that can realize proximity services, such as UWB tagless gate (UTG), thanks to centimeter-level localization accuracy based on two different ranging methods such as downlink time-difference of arrival (DL-TDoA) and double-sided two-way ranging (DS-TWR). The UTG is a UWB-based proximity service that provides a seamless gate pass system without requiring real-time mobile device (MD) tapping. The location of MD is calculated using DL-TDoA, and the MD communicates with the nearest UTG using DS-TWR to open the gate. Therefore, the knowledge about the exact location of MD is the main challenge of UTG, and hence we provide the solutions for both DL-TDoA and DS-TWR. In this paper, we propose dynamic anchor selection for extremely accurate DL-TDoA localization and pose prediction for DS-TWR, called DynaPose. The pose is defined as the actual location of MD on the human body, which affects the localization accuracy. DynaPose is based on line-of-sight (LOS) and non-LOS (NLOS) classification using deep learning for anchor selection and pose prediction. Deep learning models use the UWB channel impulse response and the inertial measurement unit embedded in the smartphone. DynaPose is implemented on Samsung Galaxy Note20 Ultra and Qorvo UWB board to show the feasibility and applicability. DynaPose achieves a LOS/NLOS classification accuracy of 0.984, 62% higher DL-TDoA localization accuracy, and ultimately detects four different poses with an accuracy of 0.961 in real-time., Comment: arXiv admin note: substantial text overlap with arXiv:2402.08399

Among the various Ultra-wideband (UWB) ranging methods, the absence of uplink communication or centralized computation makes downlink time-difference-of-arrival (DL-TDOA) localization the most suitable for large-scale industrial deployments. However, temporary or permanent obstacles in the deployment region often lead to non-line-of-sight (NLOS) channel path and signal outage effects, which result in localization errors. Prior research has addressed this problem by increasing the ranging frequency, which leads to a heavy increase in the user device power consumption. It also does not contribute to any increase in localization accuracy under line-of-sight (LOS) conditions. In this paper, we propose and implement a novel low-power channel-aware dynamic frequency DL-TDOA ranging algorithm. It comprises NLOS probability predictor based on a convolutional neural network (CNN), a dynamic ranging frequency control module, and an IMU sensor-based ranging filter. Based on the conducted experiments, we show that the proposed algorithm achieves 50% higher accuracy in NLOS conditions while having 46% lower power consumption in LOS conditions compared to baseline methods from prior research.

As commercial interest in proximity services increased, the development of various wireless localization techniques was promoted. In line with this trend, Ultra-wideband (UWB) is emerging as a promising solution that can realize proximity services thanks to centimeter-level localization accuracy. In addition, since the actual location of the mobile device (MD) on the human body, called pose, affects the localization accuracy, poses are also important to provide accurate proximity services, especially for the UWB tagless gate (UTG). In this paper, a real-time pose detector, termed D3, is proposed to estimate the pose of MD when users pass through UTG. D3 is based on line-of-sight (LOS) and non-LOS (NLOS) classification using UWB channel impulse response and utilizes the inertial measurement unit embedded in the smartphone to estimate the pose. D3 is implemented on Samsung Galaxy Note20 Ultra (i.e., SMN986B) and Qorvo UWB board to show the feasibility and applicability. D3 achieved an LOS/NLOS classification accuracy of 0.984, and ultimately detected four different poses of MD with an accuracy of 0.961 in real-time.

Shared information is a measure of mutual dependence among multiple jointly distributed random variables with finite alphabets. For a Markov chain on a tree with a given joint distribution, we give a new proof of an explicit characterization of shared information. The Markov chain on a tree is shown to possess a global Markov property based on graph separation; this property plays a key role in our proofs. When the underlying joint distribution is not known, we exploit the special form of this characterization to provide a multiarmed bandit algorithm for estimating shared information, and analyze its error performance., Comment: 13 pages, 4 figures, submitted to IEEE Transactions on Information Theory