Interview of James Kenney by Cristopher Aguilar

A 45-minute interview of Philadelphia Councilman James Kenney. Part 1 focuses on his memories of his time as a student at La Salle College. Part 2 touches upon a variety of political topics

Neural rough differential equations for long time series

Neural controlled differential equations (CDEs) are the continuous-time analogue of recurrent neural networks, as Neural ODEs are to residual networks, and offer a memory-efficient continuous-time way to model functions of potentially irregular time series. Existing methods for computing the forward pass of a Neural CDE involve embedding the incoming time series into path space, often via interpolation, and using evaluations of this path to drive the hidden state. Here, we use rough path theory to extend this formulation. Instead of directly embedding into path space, we instead represent the input signal over small time intervals through its \textit{log-signature}, which are statistics describing how the signal drives a CDE. This is the approach for solving \textit{rough differential equations} (RDEs), and correspondingly we describe our main contribution as the introduction of Neural RDEs. This extension has a purpose: by generalising the Neural CDE approach to a broader class of driving signals, we demonstrate particular advantages for tackling long time series. In this regime, we demonstrate efficacy on problems of length up to 17k observations and observe significant training speed-ups, improvements in model performance, and reduced memory requirements compared to existing approaches

New directions in the applications of rough path theory

This article provides a concise overview of some of the recent advances in the application of rough path theory to machine learning. Controlled differential equations (CDEs) are discussed as the key mathematical model to describe the interaction of a stream with a physical control system. A collection of iterated integrals known as the signature naturally arises in the description of the response produced by such interactions. The signature comes equipped with a variety of powerful properties rendering it an ideal feature map for streamed data. We summarise recent advances in the symbiosis between deep learning and CDEs, studying the link with RNNs and culminating with the Neural CDE model. We concluded with a discussion on signature kernel methods

The signature kernel is the solution of a Goursat PDE

Recently, there has been an increased interest in the development of kernel methods for learning with sequential data. The signature kernel is a learning tool with potential to handle irregularly sampled, multivariate time series. In "Kernels for sequentially ordered data" the authors introduced a kernel trick for the truncated version of this kernel avoiding the exponential complexity that would have been involved in a direct computation. Here we show that for continuously differentiable paths, the signature kernel solves a hyperbolic PDE and recognize the connection with a well known class of differential equations known in the literature as Goursat problems. This Goursat PDE only depends on the increments of the input sequences, does not require the explicit computation of signatures and can be solved efficiently using state-of-the-arthyperbolic PDE numerical solvers, giving a kernel trick for the untruncated signature kernel, with the same raw complexity as the method from "Kernels for sequentially ordered data", but with the advantage that the PDE numerical scheme is well suited for GPU parallelization, which effectively reduces the complexity by a full order of magnitude in the length of the input sequences. In addition, we extend the previous analysis to the space of geometric rough paths and establish, using classical results from rough path theory, that the rough version of the signature kernel solves a rough integral equation analogous to the aforementioned Goursat PDE. Finally, we empirically demonstrate the effectiveness of our PDE kernel as a machine learning tool in various machine learning applications dealing with sequential data. We release the library sigkernel publicly available at this https URL

The Signature Kernel is the solution of a Goursat PDE

Recently, there has been an increased interest in the development of kernel methods for learning with sequential data. The signature kernel is a learning tool with potential to handle irregularly sampled, multivariate time series. In "Kernels for sequentially ordered data" the authors introduced a kernel trick for the truncated version of this kernel avoiding the exponential complexity that would have been involved in a direct computation. Here we show that for continuously differentiable paths, the signature kernel solves a hyperbolic PDE and recognize the connection with a well known class of differential equations known in the literature as Goursat problems. This Goursat PDE only depends on the increments of the input sequences, does not require the explicit computation of signatures and can be solved efficiently using state-of-the-arthyperbolic PDE numerical solvers, giving a kernel trick for the untruncated signature kernel, with the same raw complexity as the method from "Kernels for sequentially ordered data", but with the advantage that the PDE numerical scheme is well suited for GPU parallelization, which effectively reduces the complexity by a full order of magnitude in the length of the input sequences. In addition, we extend the previous analysis to the space of geometric rough paths and establish, using classical results from rough path theory, that the rough version of the signature kernel solves a rough integral equation analogous to the aforementioned Goursat PDE. Finally, we empirically demonstrate the effectiveness of our PDE kernel as a machine learning tool in various machine learning applications dealing with sequential data. We release the library sigkernel publicly available at https://github.com/crispitagorico/sigkernel

Extreme UV QSOs

We present a sample of spectroscopically confirmed QSOs with FUV-NUV color (as measured by GALEX photometry) bluer than canonical QSO templates and than the majority of known QSOs. We analyze their FUV to NIR colors, luminosities and optical spectra. The sample includes a group of 150 objects at low redshift (z $<$ 0.5), and a group of 21 objects with redshift 1.7$<$z$<$2.6. For the low redshift objects, the "blue" FUV-NUV color may be caused by enhanced Ly$\alpha$ emission, since Ly$\alpha$ transits the GALEX FUV band from z=0.1 to z=0.47. Synthetic QSO templates constructed with Ly$\alpha$ up to 3 times stronger than in standard templates match the observed UV colors of our low redshift sample. The H$\alpha$ emission increases, and the optical spectra become bluer, with increasing absolute UV luminosity. The UV-blue QSOs at redshift about 2, where the GALEX bands sample restframe about 450-590A (FUV) and about 590-940A(NUV), are fainter than the average of UV-normal QSOs at similar redshift in NUV, while they have comparable luminosities in other bands. Therefore we speculate that their observed FUV-NUV color may be explained by a combination of steep flux rise towards short wavelengths and dust absorption below the Lyman limit, such as from small grains or crystalline carbon. The ratio of Ly$\alpha$ to CIV could be measured in 10 objects; it is higher (30% on average) than for UV-normal QSOs, and close to the value expected for shock or collisional ionization. FULL VERSION AVAILABLE FROM AUTHOR'S WEB SITE: http://dolomiti.pha.jhu.edu/papers/2009_AJ_Extreme_UV_QSOs.pdfComment: Astronomical Journal, in pres

Neural Rough Differential Equations for Long Time Series

Neural controlled differential equations (CDEs) are the continuous-time analogue of recurrent neural networks, as Neural ODEs are to residual networks, and offer a memory-efficient continuous-time way to model functions of potentially irregular time series. Existing methods for computing the forward pass of a Neural CDE involve embedding the incoming time series into path space, often via interpolation, and using evaluations of this path to drive the hidden state. Here, we use rough path theory to extend this formulation. Instead of directly embedding into path space, we instead represent the input signal over small time intervals through its \textit{log-signature}, which are statistics describing how the signal drives a CDE. This is the approach for solving \textit{rough differential equations} (RDEs), and correspondingly we describe our main contribution as the introduction of Neural RDEs. This extension has a purpose: by generalising the Neural CDE approach to a broader class of driving signals, we demonstrate particular advantages for tackling long time series. In this regime, we demonstrate efficacy on problems of length up to 17k observations and observe significant training speed-ups, improvements in model performance, and reduced memory requirements compared to existing approaches.Comment: Published at ICML 202

Detector-Free Optimization of Traffic Signal Offsets with Connected Vehicle Data

It has recently been shown that signal offset optimization is feasible using vehicle trajectory data at low levels of market penetration. This study performs offset optimization on two corridors using this type ofdata. Six weeks oftrajectory splines were processed for two corridors including 25 signalized intersections, in order to create vehicle arrival profiles, using a proposed procedure called virtual detection. After processing and filtering the data, penetration rates between 0.09-0.80% were observed, varying by approach. The arrival profiles were statistically compared against those measured with physical detectors, with the majority of the approaches showing statistically significant goodness-of-fit at a 90% confidence level. Finally, the virtual detection arrival profiles were used to optimize offsets, and compared against a solution derived from physical detector arrival profiles. The results demonstrate that virtual detection can produce good quality offsets with current market penetration rates of probe data. The study also includes a sensitivity analysis to the sample period, which shows that two weeks of data may be sufficient for data collection at current penetration rates