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1. Characteristic and Universal Tensor Product Kernels

   Szabo, Z ; Sriperumbudur, B. ( January 2018 , Journal of machine learning research)

   Maximum mean discrepancy (MMD), also called energy distance or N-distance in statistics and Hilbert-Schmidt independence criterion (HSIC), specifically distance covariance in statistics, are among the most popular and successful approaches to quantify the difference and independence of random variables, respectively. Thanks to their kernel-based foundations, MMD and HSIC are applicable on a wide variety of domains. Despite their tremendous success, quite little is known about when HSIC characterizes independence and when MMD with tensor product kernel can discriminate probability distributions. In this paper, we answer these questions by studying various notions of the characteristic property of the tensor product kernel. 

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2. Calibrated Predictive Distributions for Photometric Redshifts

   Dey, Biprateep ; Zhao, David ; Andrews, Brett ; Newman, Jeffrey ; Izbicki, Rafael ; Lee, Ann ( July 2022 , Machine Learning for Astrophysics, proceedings of the Thirty-ninth International Conference on Machine Learning (ICML 2022))

   Many astrophysical analyses depend on estimates of redshifts (a proxy for distance) determined from photometric (i.e., imaging) data alone. Inaccurate estimates of photometric redshift uncertainties can result in large systematic errors. However, probability distribution outputs from many photometric redshift methods do not follow the frequentist definition of a Probability Density Function (PDF) for redshift — i.e., the fraction of times the true redshift falls between two limits z1 and z2 should be equal to the integral of the PDF between these limits. Previous works have used the global distribution of Probability Integral Transform (PIT) values to re-calibrate PDFs, but offsetting inaccuracies in different regions of feature space can conspire to limit the efficacy of the method. We leverage a recently developed regression technique that characterizes the local PIT distribution at any location in feature space to perform a local re-calibration of photometric redshift PDFs resulting in calibrated predictive distributions. Though we focus on an example from astrophysics, our method can produce predictive distributions which are calibrated at all locations in feature space for any use case. 

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3. Re-calibrating Photometric Redshift Probability Distributions Using Feature-space Regression

   Dey, Biprateep ; Newman, Jeffrey A. ; Andrews, Brett H. ; Izbicki, Rafael ; Lee, Ann B. ; Zhao, David ; Rau, Markus Michael ; Malz, Alex I. ( December 2021 , Advances in neural information processing systems)

   Many astrophysical analyses depend on estimates of redshifts (a proxy for distance) determined from photometric (i.e., imaging) data alone. Inaccurate estimates of photometric redshift uncertainties can result in large systematic errors. However, probability distribution outputs from many photometric redshift methods do not follow the frequentist definition of a Probability Density Function (PDF) for redshift -- i.e., the fraction of times the true redshift falls between two limits z1 and z2 should be equal to the integral of the PDF between these limits. Previous works have used the global distribution of Probability Integral Transform (PIT) values to re-calibrate PDFs, but offsetting inaccuracies in different regions of feature space can conspire to limit the efficacy of the method. We leverage a recently developed regression technique that characterizes the local PIT distribution at any location in feature space to perform a local re-calibration of photometric redshift PDFs. Though we focus on an example from astrophysics, our method can produce PDFs which are calibrated at all locations in feature space for any use case. 

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4. Branching fractional Brownian motion as a model of serotonergic neurons

   Beattie-Hauser, Reece D. ; Khairnar, Gaurav R. ; House, Jonathan ; Janusonis, Skirmantas ; Metzler, Ralf ; Vojta, Thomas ( March 2023 , American Physical Society March Meeting)

   Fractional Brownian Motion (FBM) is a stochastic process with long-time correlations which has been used to model anomalous diffusion in numerous biological systems. Recently, it has been used to study the distribution of serotonergic fibers in the brain [1,2]. To better represent the biological process we are trying to simulate, we introduce the concept of branching FBM (bFBM). In this stochastic process, individual particles perform FBM but may randomly split into two. Here, we study bFBM in one space dimension in the subdiffusive and superdiffusive regimes, both in free space and on finite intervals with reflecting boundaries. We examine three possible types of behavior of the correlations (memory) at a branching event: both particles keep the memory of the previous steps, only one particle keeps the memory, and no particles keep the memory. We calculate the mean-square particle displacement, the corresponding probability distribution, and displacement correlation function. We find that the qualitative features of the bFBM process strongly depend on the type of branching event. We also discuss implications of our results for the distribution of serotonergic fibers, and we discuss possible future refinements of the model, including interactions between different fibers. [1] T. Vojta, S. Halladay, S. Skinner, S. Janusonis, T. Guggenberger, R. Metzler, Phys. Rev. E 102, 032108 (2020). [2] S. Janusonis, N. Detering, R. Metzler, T. Vojta, Front. Comput. Neurosci. 14, 56 (2020). This work was supported in part by the NSF under grant no. IIS-2112862 and by a Cottrell SEED award from Research Corporation. 

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5. Using text to teach image retrieval

   https://doi.org/10.1109/CVPRW53098.2021.00180  

   H. Dong, Z. Wang ( July 2021 , CVPR 2021 Workshop)

   Image retrieval relies heavily on the quality of the data modeling and the distance measurement in the feature space. Building on the concept of image manifold, we first propose to represent the feature space of images, learned via neural networks, as a graph. Neighborhoods in the feature space are now defined by the geodesic distance between images, represented as graph vertices or manifold samples. When limited images are available, this manifold is sparsely sampled, making the geodesic computation and the corresponding retrieval harder. To address this, we augment the manifold samples with geometrically aligned text, thereby using a plethora of sentences to teach us about images. In addition to extensive results on standard datasets illustrating the power of text to help in image retrieval, a new public dataset based on CLEVR is introduced to quantify the semantic similarity between visual data and text data. The experimental results show that the joint embedding manifold is a robust representation, allowing it to be a better basis to perform image retrieval given only an image and a textual instruction on the desired modifications over the image. 

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