Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators

In this paper we introduce a generalized Sobolev space by defining a semi-inner product formulated in terms of a vector distributional operator $\mathbf{P}$ consisting of finitely or countably many distributional operators $P_n$, which are defined on the dual space of the Schwartz space. The types of operators we consider include not only differential operators, but also more general distributional operators such as pseudo-differential operators. We deduce that a certain appropriate full-space Green function $G$ with respect to $L:=\mathbf{P}^{\ast T}\mathbf{P}$ now becomes a conditionally positive definite function. In order to support this claim we ensure that the distributional adjoint operator $\mathbf{P}^{\ast}$ of $\mathbf{P}$ is well-defined in the distributional sense. Under sufficient conditions, the native space (reproducing-kernel Hilbert space) associated with the Green function $G$ can be isometrically embedded into or even be isometrically equivalent to a generalized Sobolev space. As an application, we take linear combinations of translates of the Green function with possibly added polynomial terms and construct a multivariate minimum-norm interpolant $s_{f,X}$ to data values sampled from an unknown generalized Sobolev function $f$ at data sites located in some set $X \subset \mathbb{R}^d$. We provide several examples, such as Mat\'ern kernels or Gaussian kernels, that illustrate how many reproducing-kernel Hilbert spaces of well-known reproducing kernels are isometrically equivalent to a generalized Sobolev space. These examples further illustrate how we can rescale the Sobolev spaces by the vector distributional operator $\mathbf{P}$. Introducing the notion of scale as part of the definition of a generalized Sobolev space may help us to choose the "best" kernel function for kernel-based approximation methods.Comment: Update version of the publish at Num. Math. closed to Qi Ye's Ph.D. thesis (\url{http://mypages.iit.edu/~qye3/PhdThesis-2012-AMS-QiYe-IIT.pdf}

The Deformable Mirror Demonstration Mission (DeMi) CubeSat: optomechanical design validation and laboratory calibration

Coronagraphs on future space telescopes will require precise wavefront correction to detect Earth-like exoplanets near their host stars. High-actuator count microelectromechanical system (MEMS) deformable mirrors provide wavefront control with low size, weight, and power. The Deformable Mirror Demonstration Mission (DeMi) payload will demonstrate a 140 actuator MEMS deformable mirror (DM) with \SI{5.5}{\micro\meter} maximum stroke. We present the flight optomechanical design, lab tests of the flight wavefront sensor and wavefront reconstructor, and simulations of closed-loop control of wavefront aberrations. We also present the compact flight DM controller, capable of driving up to 192 actuator channels at 0-250V with 14-bit resolution. Two embedded Raspberry Pi 3 compute modules are used for task management and wavefront reconstruction. The spacecraft is a 6U CubeSat (30 cm x 20 cm x 10 cm) and launch is planned for 2019.Comment: 15 pages, 10 figues. Presented at SPIE Astronomical Telescopes + Instrumentation, Austin, Texas, US

Solving Support Vector Machines in Reproducing Kernel Banach Spaces with Positive Definite Functions

In this paper we solve support vector machines in reproducing kernel Banach spaces with reproducing kernels defined on nonsymmetric domains instead of the traditional methods in reproducing kernel Hilbert spaces. Using the orthogonality of semi-inner-products, we can obtain the explicit representations of the dual (normalized-duality-mapping) elements of support vector machine solutions. In addition, we can introduce the reproduction property in a generalized native space by Fourier transform techniques such that it becomes a reproducing kernel Banach space, which can be even embedded into Sobolev spaces, and its reproducing kernel is set up by the related positive definite function. The representations of the optimal solutions of support vector machines (regularized empirical risks) in these reproducing kernel Banach spaces are formulated explicitly in terms of positive definite functions, and their finite numbers of coefficients can be computed by fixed point iteration. We also give some typical examples of reproducing kernel Banach spaces induced by Mat\'ern functions (Sobolev splines) so that their support vector machine solutions are well computable as the classical algorithms. Moreover, each of their reproducing bases includes information from multiple training data points. The concept of reproducing kernel Banach spaces offers us a new numerical tool for solving support vector machines.Comment: 26 page

Localness of energy cascade in hydrodynamic turbulence, II. Sharp spectral filter

We investigate the scale-locality of subgrid-scale (SGS) energy flux and inter-band energy transfers defined by the sharp spectral filter. We show by rigorous bounds, physical arguments and numerical simulations that the spectral SGS flux is dominated by local triadic interactions in an extended turbulent inertial-range. Inter-band energy transfers are also shown to be dominated by local triads if the spectral bands have constant width on a logarithmic scale. We disprove in particular an alternative picture of ``local transfer by nonlocal triads,'' with the advecting wavenumber mode at the energy peak. Although such triads have the largest transfer rates of all {\it individual} wavenumber triads, we show rigorously that, due to their restricted number, they make an asymptotically negligible contribution to energy flux and log-banded energy transfers at high wavenumbers in the inertial-range. We show that it is only the aggregate effect of a geometrically increasing number of local wavenumber triads which can sustain an energy cascade to small scales. Furthermore, non-local triads are argued to contribute even less to the space-average energy flux than is implied by our rigorous bounds, because of additional cancellations from scale-decorrelation effects. We can thus recover the -4/3 scaling of nonlocal contributions to spectral energy flux predicted by Kraichnan's ALHDIA and TFM closures. We support our results with numerical data from a $512^3$ pseudospectral simulation of isotropic turbulence with phase-shift dealiasing. We conclude that the sharp spectral filter has a firm theoretical basis for use in large-eddy simulation (LES) modeling of turbulent flows.Comment: 42 pages, 9 figure

Electrophysiological Signatures of Spatial Boundaries in the Human Subiculum.

Environmental boundaries play a crucial role in spatial navigation and memory across a wide range of distantly related species. In rodents, boundary representations have been identified at the single-cell level in the subiculum and entorhinal cortex of the hippocampal formation. Although studies of hippocampal function and spatial behavior suggest that similar representations might exist in humans, boundary-related neural activity has not been identified electrophysiologically in humans until now. To address this gap in the literature, we analyzed intracranial recordings from the hippocampal formation of surgical epilepsy patients (of both sexes) while they performed a virtual spatial navigation task and compared the power in three frequency bands (1-4, 4-10, and 30-90 Hz) for target locations near and far from the environmental boundaries. Our results suggest that encoding locations near boundaries elicited stronger theta oscillations than for target locations near the center of the environment and that this difference cannot be explained by variables such as trial length, speed, movement, or performance. These findings provide direct evidence of boundary-dependent neural activity localized in humans to the subiculum, the homolog of the hippocampal subregion in which most boundary cells are found in rodents, and indicate that this system can represent attended locations that rather than the position of one\u27s own body

Essential Role for endogenous siRNAs during meiosis in mouse oocytes.

The RNase III enzyme DICER generates both microRNAs (miRNAs) and endogenous short interfering RNAs (endo-siRNAs). Both small RNA species silence gene expression post-transcriptionally in association with the ARGONAUTE (AGO) family of proteins. In mammals, there are four AGO proteins (AGO1-4), of which only AGO2 possesses endonucleolytic activity. siRNAs trigger endonucleolytic cleavage of target mRNAs, mediated by AGO2, whereas miRNAs cause translational repression and mRNA decay through association with any of the four AGO proteins. Dicer deletion in mouse oocytes leads to female infertility due to defects during meiosis I. Because mouse oocytes express both miRNAs and endo-siRNAs, this phenotype could be due to the absence of either class of small RNA, or both. However, we and others demonstrated that miRNA function is suppressed in mouse oocytes, which suggested that endo-siRNAs, not miRNAs, are essential for female meiosis. To determine if this was the case we generated mice that express a catalytically inactive knock-in allele of Ago2 (Ago2ADH) exclusively in oocytes and thereby disrupted the function of siRNAs. Oogenesis and hormonal response are normal in Ago2ADH oocytes, but meiotic maturation is impaired, with severe defects in spindle formation and chromosome alignment that lead to meiotic catastrophe. The transcriptome of these oocytes is widely perturbed and shows a highly significant correlation with the transcriptome of Dicer null and Ago2 null oocytes. Expression of the mouse transcript (MT), the most abundant transposable element in mouse oocytes, is increased. This study reveals that endo-siRNAs are essential during meiosis I in mouse females, demonstrating a role for endo-siRNAs in mammals.This research was supported by the National Institutes of Health Grants HD022681 (to RMS), and R37 GM062534-14 (to GJH), National Human Genome Research Institute 5T32HG000046-13 (to FL) and by a kind gift from Kathryn W. Davis. GJH is an investigator of the Howard Hughes Medical Institute. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.This is the final version of the article. It first appeared from PLoS via http://dx.doi.org/10.1371/journal.pgen.100501

Data-Efficient Policy Selection for Navigation in Partial Maps via Subgoal-Based Abstraction

We present a novel approach for fast and reliable policy selection for navigation in partial maps. Leveraging the recent learning-augmented model-based Learning over Subgoals Planning (LSP) abstraction to plan, our robot reuses data collected during navigation to evaluate how well other alternative policies could have performed via a procedure we call offline alt-policy replay. Costs from offline alt-policy replay constrain policy selection among the LSP-based policies during deployment, allowing for improvements in convergence speed, cumulative regret and average navigation cost. With only limited prior knowledge about the nature of unseen environments, we achieve at least 67% and as much as 96% improvements on cumulative regret over the baseline bandit approach in our experiments in simulated maze and office-like environments.Comment: 8 pages, 5 figure

Learning Augmented, Multi-Robot Long-Horizon Navigation in Partially Mapped Environments

We present a novel approach for efficient and reliable goal-directed long-horizon navigation for a multi-robot team in a structured, unknown environment by predicting statistics of unknown space. Building on recent work in learning-augmented model based planning under uncertainty, we introduce a high-level state and action abstraction that lets us approximate the challenging Dec-POMDP into a tractable stochastic MDP. Our Multi-Robot Learning over Subgoals Planner (MR-LSP) guides agents towards coordinated exploration of regions more likely to reach the unseen goal. We demonstrate improvement in cost against other multi-robot strategies; in simulated office-like environments, we show that our approach saves 13.29% (2 robot) and 4.6% (3 robot) average cost versus standard non-learned optimistic planning and a learning-informed baseline.Comment: 7 pages, 7 figures, ICRA202

Ripple oscillations in the left temporal neocortex are associated with impaired verbal episodic memory encoding

Background: We sought to determine if ripple oscillations (80-120Hz), detected in intracranial EEG (iEEG) recordings of epilepsy patients, correlate with an enhancement or disruption of verbal episodic memory encoding. Methods: We defined ripple and spike events in depth iEEG recordings during list learning in 107 patients with focal epilepsy. We used logistic regression models (LRMs) to investigate the relationship between the occurrence of ripple and spike events during word presentation and the odds of successful word recall following a distractor epoch, and included the seizure onset zone (SOZ) as a covariate in the LRMs. Results: We detected events during 58,312 word presentation trials from 7,630 unique electrode sites. The probability of ripple on spike (RonS) events was increased in the seizure onset zone (SOZ, p<0.04). In the left temporal neocortex RonS events during word presentation corresponded with a decrease in the odds ratio (OR) of successful recall, however this effect only met significance in the SOZ (OR of word recall 0.71, 95% CI: 0.59-0.85, n=158 events, adaptive Hochberg p<0.01). Ripple on oscillation events (RonO) that occurred in the left temporal neocortex non-SOZ also correlated with decreased odds of successful recall (OR 0.52, 95% CI: 0.34-0.80, n=140, adaptive Hochberg , p<0.01). Spikes and RonS that occurred during word presentation in the left middle temporal gyrus during word presentation correlated with the most significant decrease in the odds of successful recall, irrespective of the location of the SOZ (adaptive Hochberg, p<0.01). Conclusion: Ripples and spikes generated in left temporal neocortex are associated with impaired verbal episodic memory encoding

Active Information Gathering for Long-Horizon Navigation Under Uncertainty by Learning the Value of Information

We address the task of long-horizon navigation in partially mapped environments for which active gathering of information about faraway unseen space is essential for good behavior. We present a novel planning strategy that, at training time, affords tractable computation of the value of information associated with revealing potentially informative regions of unseen space, data used to train a graph neural network to predict the goodness of temporally-extended exploratory actions. Our learning-augmented model-based planning approach predicts the expected value of information of revealing unseen space and is capable of using these predictions to actively seek information and so improve long-horizon navigation. Across two simulated office-like environments, our planner outperforms competitive learned and non-learned baseline navigation strategies, achieving improvements of up to 63.76% and 36.68%, demonstrating its capacity to actively seek performance-critical information.Comment: Submitted at IROS'24. arXiv admin note: text overlap with arXiv:2307.1450