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The principal ratio of a graph is the ratio of the greatest and least entry of its principal eigenvector. Since the principal ratio compares the extreme values of the principal eigenvector it is sensitive to outliers. This can be problematic for graphs (networks) drawn from empirical data. To account for this we consider the dispersion of the principal eigenvector (and degree vector). More precisely, we consider the coefficient of variation of the aforementioned vectors, that is, the ratio of the vector's standard deviation and mean. We show how both of these statistics are bounded above by the same function of the principal ratio. Further this bound is sharp for regular graphs. The goal of this paper is to show that the coefficient of variation of the principal eigenvector (and degree vector) can converge or diverge to the principal ratio in the limit. In doing so we find an example of a graph family (the complete split graph) whose principal ratio converges to the golden ratio. We conclude with conjectures concerning extremal graphs of the aforementioned statistics and interesting properties of the complete split graph., Comment: 21 pages, 4 figures, 1 table

Graphs (i.e., networks) have become an integral tool for the representation and analysis of relational data. Advances in data gathering have lead to multi-relational data sets which exhibit greater depth and scope. In certain cases, this data can be modeled using a hypergraph. However, in practice analysts typically reduce the dimensionality of the data (whether consciously or otherwise) to accommodate a traditional graph model. In recent years spectral hypergraph theory has emerged to study the eigenpairs of the adjacency hypermatrix of a uniform hypergraph. We show how analyzing multi-relational data, via a hypermatrix associated to the aforementioned hypergraph, can lead to conclusions different from those when the data is projected down to its co-occurrence matrix. In particular, we provide an example of a uniform hypergraph where the most central vertex (\`a la eigencentrality) changes depending on the order of the associated matrix. To the best of our knowledge this is the first known hypergraph to exhibit this property., Comment: 13 pages, 4 figures, 2 tables

This paper introduces deep synoptic Monte Carlo planning (DSMCP) for large imperfect information games. The algorithm constructs a belief state with an unweighted particle filter and plans via playouts that start at samples drawn from the belief state. The algorithm accounts for uncertainty by performing inference on "synopses," a novel stochastic abstraction of information states. DSMCP is the basis of the program Penumbra, which won the official 2020 reconnaissance blind chess competition versus 33 other programs. This paper also evaluates algorithm variants that incorporate caution, paranoia, and a novel bandit algorithm. Furthermore, it audits the synopsis features used in Penumbra with per-bit saliency statistics., Comment: Accepted to NeurIPS 2021

The Harary-Sachs theorem for $k$-uniform hypergraphs equates the codegree-$d$ coefficient of the adjacency characteristic polynomial of a uniform hypergraph with a weighted sum of subgraph counts over certain multi-hypergraphs with $d$ edges. We begin by showing that the classical Harary-Sachs theorem for graphs is indeed a special case of this general theorem. To this end we apply the generalized Harary-Sachs theorem to the leading coefficients of the characteristic polynomial of various hypergraphs. In particular, we provide explicit and asymptotic formulas for the contribution of the $k$-uniform simplex to the codegree-$d$ coefficient. Moreover, we provide an explicit formula for the leading terms of the characteristic polynomial of a 3-uniform hypergraph and further show how this can be used to determine the complete spectrum of a hypergraph. We conclude with a conjecture concerning the multiplicity of the zero-eigenvalue of a hypergraph., Comment: arXiv admin note: substantial text overlap with arXiv:1812.00468

Consider a graph $G$ where each vertex is visibly labelled as a member of a distinct class, but also has a hidden binary state: wild or tame. Edges with end points in the same class are called agreement edges. Premise: an edge connecting vertices in different classes -- a conflict edge -- is allowed only when at least one end point is wild. Interpret wild status as readiness to form connections with any other vertex, regardless of class -- a form of class disaffiliation. The learning goal is to classify each vertex as wild or tame using its neighborhood data. In applications such as communications metadata, bio-informatics, retailing, or bibliography, adjacency in $G$ is typically created by paths of length two in a transactional bipartite graph $B$. Class labelling, imported from a reference data source, is typically assortative, so agreement edges predominate. Conflict edges represent observed behavior (from $B$) inconsistent with prior labelling of $V(G)$. Wild vertices are those whose label is uninformative. The hidden ancestor graph constitutes a natural model for generating agreement edges and conflict edges, depending on a latent tree structure. The model is able to manifest high clustering rates and heavy-tailed degree distributions typical of social and spatial networks. It can be fitted to graph data using a few measurable graph parameters, and supplies a natural statistical classifier for wild versus tame., Comment: 35 pages, 12 figures

There has been clamor and dismay at the expected future economic and social costs for children affected by school closures from COVID-19. A New York Times editorial, for example, stated: "A learning reversal of this magnitude could hobble an entire generation." However, the nearly universal belief in the substantial social and economic benefits of education faces some disquieting counterevidence. School closings will likely have little social cost. One piece of counterevidence is the effects of England's extensions of compulsory schooling in 1919-1922, 1947, and 1972. This shows compellingly that more schooling for students of lower socioeconomic status produced little or no gains in incomes or in broader measures of living standards, such as longevity. In their research, the authors exploit some new large-scale sources to examine the social and economic effects of the 1947 and 1972 schooling extensions and to also measure the effects of the 1919-1922 extension. The analysis of the effects of the 1947 schooling extension, by quarter of birth for the affected cohorts, finds that there is no sign of the 5 percent increase in permanent income that the schooling literature would predict. Indeed, the point estimate is for no gain at all. The picture is the same for the 1972 extension. If the effects of extensions of compulsory schooling in England in the 20th century are a guide for the present, lost schooling will not have any long-term social costs. [This research brief is based on Gregory Clark and Neil Cummins, "Does Education Matter? Tests from Extensions of Compulsory Schooling in England and Wales 1919-22, 1947, and 1972," Center for Economic and Policy Research Discussion Paper no. 15252, September 2020.]

The popularity and widespread use of pruning and quantization is driven by the severe resource constraints of deploying deep neural networks to environments with strict latency, memory and energy requirements. These techniques achieve high levels of compression with negligible impact on top-line metrics (top-1 and top-5 accuracy). However, overall accuracy hides disproportionately high errors on a small subset of examples; we call this subset Compression Identified Exemplars (CIE). We further establish that for CIE examples, compression amplifies existing algorithmic bias. Pruning disproportionately impacts performance on underrepresented features, which often coincides with considerations of fairness. Given that CIE is a relatively small subset but a great contributor of error in the model, we propose its use as a human-in-the-loop auditing tool to surface a tractable subset of the dataset for further inspection or annotation by a domain expert. We provide qualitative and quantitative support that CIE surfaces the most challenging examples in the data distribution for human-in-the-loop auditing.

Objective: This paper aims to demonstrate functional discriminability among restored hand sensations with different locations, qualities, and intensities that are evoked by microelectrode stimulation of residual afferent fibers in human amputees. Methods: We implanted a Utah Slanted Electrode Array (USEA) in the median and ulnar residual arm nerves of three transradial amputees and delivered stimulation via different electrodes and at different frequencies to produce various locations, qualities, and intensities of sensation on the missing hand. Blind discrimination trials were performed to determine how well subjects could discriminate among these restored sensations. Results: Subjects discriminated among restored sensory percepts with varying cutaneous and proprioceptive locations, qualities, and intensities in blind trials, including discrimination among up to 10 different location-intensity combinations (15/30 successes, p < 0.0005). Variations in the site of stimulation within the nerve, via electrode selection, enabled discrimination among up to 5 locations and qualities (35/35 successes, p < 0.0001). Variations in the stimulation frequency enabled discrimination among 4 different intensities at the same location (13/20 successes, p < 0.005). One subject discriminated among simultaneous, alternating, and isolated stimulation of two different USEA electrodes, as may be desired during multi-sensor closed-loop prosthesis use (20/25 successes, p < 0.001). Conclusion: USEA stimulation enables encoding of a diversity of functionally discriminable sensations with different locations, qualities, and intensities. Significance: These percepts provide a potentially rich source of sensory feedback that may enhance performance and embodiment during multi-sensor, closed-loop prosthesis use., Comment: 19 pages

The dexterity of conventional myoelectric prostheses is limited in part by the small datasets used to train the control algorithms. Variations in surface electrode positioning make it difficult to collect consistent data and to estimate motor intent reliably over time. To address these challenges, we developed an inexpensive, easy-to-don sleeve that can record robust and repeatable surface electromyography from 32 embedded monopolar electrodes. Embedded grommets are used to consistently align the sleeve with natural skin markings (e.g., moles, freckles, scars). The sleeve can be manufactured in a few hours for less than $60. Data from seven intact participants show the sleeve provides a signal-to-noise ratio of 14, a don-time under 11 seconds, and sub-centimeter precision for electrode placement. Furthermore, in a case study with one intact participant, we use the sleeve to demonstrate that neural networks can provide simultaneous and proportional control of six degrees of freedom, even 263 days after initial algorithm training. We also highlight that consistent recordings, accumulated over time to establish a large dataset, significantly improve dexterity. These results suggest that deep learning with a 74-layer neural network can substantially improve the dexterity and stability of myoelectric prosthetic control, and that deep-learning techniques can be readily instantiated and further validated through inexpensive sleeves/sockets with consistent recording locations., Comment: MEC2020

Intuitive control of prostheses relies on training algorithms to correlate biological recordings to motor intent. The quality of the training dataset is critical to run-time performance, but it is difficult to label hand kinematics accurately after the hand has been amputated. We quantified the accuracy and precision of labeling hand kinematics for two different approaches: 1) assuming a participant is perfectly mimicking predetermined motions of a prosthesis (mimicked training), and 2) assuming a participant is perfectly mirroring their contralateral hand during identical bilateral movements (mirrored training). We compared these approaches in non-amputee individuals, using an infrared camera to track eight different joint angles of the hands in real-time. Aggregate data revealed that mimicked training does not account for biomechanical coupling or temporal changes in hand posture. Mirrored training was significantly more accurate and precise at labeling hand kinematics. However, when training a modified Kalman filter to estimate motor intent, the mimicked and mirrored training approaches were not significantly different. The results suggest that the mirrored training approach creates a more faithful but more complex dataset. Advanced algorithms, more capable of learning the complex mirrored training dataset, may yield better run-time prosthetic control., Comment: IEEE EMBC 2020

Electrical stimulation of residual nerves can be used to provide amputees with intuitive sensory feedback. An important aspect of this artificial sensory feedback is the ability to convey the magnitude of tactile stimuli. Using classical psychophysical methods, we quantified the just-noticeable differences for electrocutaneous stimulation pulse frequency in both intact participants and one transradial amputee. For the transradial amputee, we also quantified the just-noticeable difference of intraneural microstimulation pulse frequency via chronically implanted Utah Slanted Electrode Arrays. We demonstrate that intensity discrimination is similar across conditions: intraneural microstimulation of the residual nerves, electrocutaneous stimulation of the reinnervated skin on the residual limb, and electrocutaneous stimulation of intact hands. We also show that intensity discrimination performance is significantly better at lower pulse frequencies than at higher ones - a finding that's unique to electrocutaneous and intraneural stimulation and suggests that supplemental sensory cues may be present at lower pulse frequencies. These results can help guide the implementation of artificial sensory feedback for sensorized bionic arms., Comment: IEEE EMBC 2020

Multiarticulate bionic arms are now capable of mimicking the endogenous movements of the human hand. 3D-printing has reduced the cost of prosthetic hands themselves, but there is currently no low-cost alternative to dexterous electromyographic (EMG) control systems. To address this need, we developed an inexpensive (~$675) and portable EMG control system by integrating low-cost microcontrollers with an EMG acquisition device. We validated signal acquisition by comparing the signal-to-noise ratio (SNR) of our system with that of a high-end research-grade system. We also demonstrate the ability to use the low-cost control system for proportional and independent control of various prosthetic hands in real-time. We found that the SNR of the low-cost control system was statistically no worse than 44% of the SNR of a research-grade control system. The RMSEs of predicted hand movements (from a modified Kalman filter) were typically a few percent better than, and not more than 6% worse than, RMSEs of a research-grade system for up to six degrees of freedom when only relatively few (six) EMG electrodes were used. However, RMSEs were generally higher than RMSEs of research-grade systems that utilize considerably more (32) EMG electrodes, guiding future work towards increasing electrode count. Successful instantiation of this low-cost control system constitutes an important step towards the commercialization and wide-spread availability of dexterous bionic hands., Comment: IEEE EMBC 2020

Deep neural network pruning and quantization techniques have demonstrated it is possible to achieve high levels of compression with surprisingly little degradation to test set accuracy. However, this measure of performance conceals significant differences in how different classes and images are impacted by model compression techniques. We find that models with radically different numbers of weights have comparable top-line performance metrics but diverge considerably in behavior on a narrow subset of the dataset. This small subset of data points, which we term Pruning Identified Exemplars (PIEs) are systematically more impacted by the introduction of sparsity. Compression disproportionately impacts model performance on the underrepresented long-tail of the data distribution. PIEs over-index on atypical or noisy images that are far more challenging for both humans and algorithms to classify. Our work provides intuition into the role of capacity in deep neural networks and the trade-offs incurred by compression. An understanding of this disparate impact is critical given the widespread deployment of compressed models in the wild.

Bypass sockets allow researchers to perform tests of prosthetic systems from the prosthetic user's perspective. We designed a modular upper-limb bypass socket with 3D-printed components that can be easily modified for use with a variety of terminal devices. Our bypass socket preserves access to forearm musculature and the hand, which are necessary for surface electromyography and to provide substituted sensory feedback. Our bypass socket allows a sufficient range of motion to complete tasks in the frontal working area, as measured on non-amputee participants. We examined the performance of non-amputee participants using the bypass socket on the original and modified Box and Block Tests. Participants moved 11.3 +/- 2.7 and 11.7 +/- 2.4 blocks in the original and modified Box and Block Tests (mean +/- SD), respectively, within the range of reported scores using amputee participants. Range-of-motion for users wearing the bypass socket meets or exceeds most reported range-of-motion requirements for activities of daily living. The bypass socket was originally designed with a freely rotating wrist; we found that adding elastic resistance to user wrist rotation while wearing the bypass socket had no significant effect on motor decode performance. We have open-sourced the design files and an assembly manual for the bypass socket. We anticipate that the bypass socket will be a useful tool to evaluate and develop sensorized myoelectric prosthesis technology., Comment: 8 pages, 5 figures

Background: Multi-articulate prostheses are capable of performing dexterous hand movements. However, clinically available control strategies fail to provide users with intuitive, independent and proportional control over multiple degrees of freedom (DOFs) in real-time. New Method: We detail the use of a modified Kalman filter (MKF) to provide intuitive, independent and proportional control over six-DOF prostheses such as the DEKA "LUKE" Arm. Input features include neural firing rates recorded from Utah Slanted Electrode Arrays and mean absolute value of intramuscular electromyographic (EMG) recordings. Ad-hoc modifications include thresholds and non-unity gains on the output of a Kalman filter. Results: We demonstrate that both neural and EMG data can be combined effectively. We also highlight that modifications can be optimized to significantly improve performance relative to an unmodified Kalman filter. Thresholds significantly reduced unintended movement and promoted more independent control of the different DOFs. Gain were significantly greater than one and served to amplify participant effort. Optimal modifications can be determined quickly offline and translate to functional improvements online. Using a portable take-home system, participants performed various activities of daily living. Comparison with Existing Methods: In contrast to pattern recognition, the MKF allows users to continuously modulate their force output, which is critical for fine dexterity. The MKF is also computationally efficient and can be trained in less than five minutes. Conclusions: The MKF can be used to explore the functional and psychological benefits associated with long-term, at-home control of dexterous prosthetic hands., Comment: 10 figures. Accepted in J. Neurosci. Methods (2019)

We generalize the Harary-Sachs theorem to $k$-uniform hypergraphs: the codegree-$d$ coefficient of the characteristic polynomial of a uniform hypergraph ${\cal H}$ can be expressed as a weighted sum of subgraph counts over certain multi-hypergraphs with $d$ edges. We include a detailed description of the aforementioned multi-hypergraphs and a formula for their corresponding weights.

The crossing number ${\mbox {cr}}(G)$ of a graph $G=(V,E)$ is the smallest number of edge crossings over all drawings of $G$ in the plane. For any $k\ge 1$, the $k$-planar crossing number of $G$, ${\mbox {cr}}_k(G)$, is defined as the minimum of ${\mbox {cr}}(G_1)+{\mbox {cr}}(G_2)+\ldots+{\mbox {cr}}(G_{k})$ over all graphs $G_1, G_2,\ldots, G_{k}$ with $\cup_{i=1}^{k}G_i=G$. Pach et al. [\emph{Computational Geometry: Theory and Applications} {\bf 68} 2--6, (2018)] showed that for every $k\ge 1$, we have ${\mbox {cr}}_k(G)\le \left(\frac{2}{k^2}-\frac1{k^3}\right){\mbox {cr}}(G)$ and that this bound does not remain true if we replace the constant $\frac{2}{k^2}-\frac1{k^3}$ by any number smaller than $\frac1{k^2}$. We improve the upper bound to $\frac{1}{k^2}(1+o(1))$ as $k\rightarrow \infty$. For the class of bipartite graphs, we show that the best constant is exactly $\frac{1}{k^2}$ for every $k$. The results extend to the rectilinear variant of the $k$-planar crossing number., Comment: 13 pages, 1 figure, 1 table

We show that a monic univariate polynomial over a field of characteristic zero, with $k$ distinct non-zero known roots, is determined by its $k$ proper leading coefficients by providing an explicit algorithm for computing the multiplicities of each root. We provide a version of the result and accompanying algorithm when the field is not algebraically closed by considering the minimal polynomials of the roots. Furthermore, we show how to perform the aforementioned algorithm in a numerically stable manner over $\mathbb{C}$, and then apply it to obtain new characteristic polynomials of hypergraphs.

We extend the results of Zhang et al. to show that $\lambda$ is an eigenvalue of a $k$-uniform hypertree $(k \geq 3)$ if and only if it is a root of a particular matching polynomial for a connected induced subtree. We then use this to provide a spectral characterization for power hypertrees. Notably, the situation is quite different from that of ordinary trees, i.e., $2$-uniform trees. We conclude by presenting an example (an $11$ vertex, $3$-uniform non-power hypertree) illustrating these phenomena.

In this note we provide an improved upper bound on the biplanar crossing number of the 8-dimensional hypercube. The $k$-planar crossing number of a graph $cr_k(G)$ is the number of crossings required when every edge of $G$ must be drawn in one of $k$ distinct planes. It was shown in Czabarka et al. that $cr_2(Q_8) \leq 256$ which we improve to $cr_2(Q_8) \leq 128$. Our approach highlights the relationship between symmetric drawings and the study of $k$-planar crossing numbers. We conclude with several open questions concerning this relationship.