arXiv subjects

Jaehoon Lee

Publications and source records attributed to Jaehoon Lee.

At least 73 records · Page 4Linked to original sources

arXiv2022

Time Series Forecasting with Hypernetworks Generating Parameters in Advance

Forecasting future outcomes from recent time series data is not easy, especially when the future data are different from the past (i.e. time series are under temporal drifts). Existing approaches show limited performances under data drifts, and we identify the main reason: It takes time for a model to collect sufficient training data and adjust its parameters for complicated temporal patterns whenever the underlying dynamics change. To address this issue, we study a new approach; instead of adjusting model parameters (by continuously re-training a model on new data), we build a hypernetwork that generates other target models' parameters expected to perform well on the future data. Therefore, we can adjust the model parameters beforehand (if the hypernetwork is correct). We conduct extensive experiments with 6 target models, 6 baselines, and 4 datasets, and show that our HyperGPA outperforms other baselines.

cs.LG↗

arXiv2022

Frankel's property for free boundary minimal hypersurfaces in the Riemannian Schwarzschild manifolds

We study the behavior of minimal hypersurfaces in the Schwarzschild $n$-manifolds that intersect the horizon orthogonally along the boundary. We show that a free boundary minimal hypersurface and a totally geodesic hyperplane must intersect when the distance between them is achieved in a bounded region. We also discuss when the Schwarzschild metric is perturbed in a way that its scalar curvature is no longer positive.

math.DG↗

arXiv2022

EXIT: Extrapolation and Interpolation-based Neural Controlled Differential Equations for Time-series Classification and Forecasting

Deep learning inspired by differential equations is a recent research trend and has marked the state of the art performance for many machine learning tasks. Among them, time-series modeling with neural controlled differential equations (NCDEs) is considered as a breakthrough. In many cases, NCDE-based models not only provide better accuracy than recurrent neural networks (RNNs) but also make it possible to process irregular time-series. In this work, we enhance NCDEs by redesigning their core part, i.e., generating a continuous path from a discrete time-series input. NCDEs typically use interpolation algorithms to convert discrete time-series samples to continuous paths. However, we propose to i) generate another latent continuous path using an encoder-decoder architecture, which corresponds to the interpolation process of NCDEs, i.e., our neural network-based interpolation vs. the existing explicit interpolation, and ii) exploit the generative characteristic of the decoder, i.e., extrapolation beyond the time domain of original data if needed. Therefore, our NCDE design can use both the interpolated and the extrapolated information for downstream machine learning tasks. In our experiments with 5 real-world datasets and 12 baselines, our extrapolation and interpolation-based NCDEs outperform existing baselines by non-trivial margins.

cs.LG↗

arXiv2022

Fast Neural Kernel Embeddings for General Activations

Infinite width limit has shed light on generalization and optimization aspects of deep learning by establishing connections between neural networks and kernel methods. Despite their importance, the utility of these kernel methods was limited in large-scale learning settings due to their (super-)quadratic runtime and memory complexities. Moreover, most prior works on neural kernels have focused on the ReLU activation, mainly due to its popularity but also due to the difficulty of computing such kernels for general activations. In this work, we overcome such difficulties by providing methods to work with general activations. First, we compile and expand the list of activation functions admitting exact dual activation expressions to compute neural kernels. When the exact computation is unknown, we present methods to effectively approximate them. We propose a fast sketching method that approximates any multi-layered Neural Network Gaussian Process (NNGP) kernel and Neural Tangent Kernel (NTK) matrices for a wide range of activation functions, going beyond the commonly analyzed ReLU activation. This is done by showing how to approximate the neural kernels using the truncated Hermite expansion of any desired activation functions. While most prior works require data points on the unit sphere, our methods do not suffer from such limitations and are applicable to any dataset of points in $\mathbb{R}^d$. Furthermore, we provide a subspace embedding for NNGP and NTK matrices with near input-sparsity runtime and near-optimal target dimension which applies to any \emph{homogeneous} dual activation functions with rapidly convergent Taylor expansion. Empirically, with respect to exact convolutional NTK (CNTK) computation, our method achieves $106\times$ speedup for approximate CNTK of a 5-layer Myrtle network on CIFAR-10 dataset.

cs.LG↗

arXiv2022

LORD: Lower-Dimensional Embedding of Log-Signature in Neural Rough Differential Equations

The problem of processing very long time-series data (e.g., a length of more than 10,000) is a long-standing research problem in machine learning. Recently, one breakthrough, called neural rough differential equations (NRDEs), has been proposed and has shown that it is able to process such data. Their main concept is to use the log-signature transform, which is known to be more efficient than the Fourier transform for irregular long time-series, to convert a very long time-series sample into a relatively shorter series of feature vectors. However, the log-signature transform causes non-trivial spatial overheads. To this end, we present the method of LOweR-Dimensional embedding of log-signature (LORD), where we define an NRDE-based autoencoder to implant the higher-depth log-signature knowledge into the lower-depth log-signature. We show that the encoder successfully combines the higher-depth and the lower-depth log-signature knowledge, which greatly stabilizes the training process and increases the model accuracy. In our experiments with benchmark datasets, the improvement ratio by our method is up to 75\% in terms of various classification and forecasting evaluation metrics.

cs.LG↗

arXiv2022

Invertible Tabular GANs: Killing Two Birds with OneStone for Tabular Data Synthesis

Tabular data synthesis has received wide attention in the literature. This is because available data is often limited, incomplete, or cannot be obtained easily, and data privacy is becoming increasingly important. In this work, we present a generalized GAN framework for tabular synthesis, which combines the adversarial training of GANs and the negative log-density regularization of invertible neural networks. The proposed framework can be used for two distinctive objectives. First, we can further improve the synthesis quality, by decreasing the negative log-density of real records in the process of adversarial training. On the other hand, by increasing the negative log-density of real records, realistic fake records can be synthesized in a way that they are not too much close to real records and reduce the chance of potential information leakage. We conduct experiments with real-world datasets for classification, regression, and privacy attacks. In general, the proposed method demonstrates the best synthesis quality (in terms of task-oriented evaluation metrics, e.g., F1) when decreasing the negative log-density during the adversarial training. If increasing the negative log-density, our experimental results show that the distance between real and fake records increases, enhancing robustness against privacy attacks.

cs.LG↗

arXiv2022

Dataset Distillation with Infinitely Wide Convolutional Networks

The effectiveness of machine learning algorithms arises from being able to extract useful features from large amounts of data. As model and dataset sizes increase, dataset distillation methods that compress large datasets into significantly smaller yet highly performant ones will become valuable in terms of training efficiency and useful feature extraction. To that end, we apply a novel distributed kernel based meta-learning framework to achieve state-of-the-art results for dataset distillation using infinitely wide convolutional neural networks. For instance, using only 10 datapoints (0.02% of original dataset), we obtain over 65% test accuracy on CIFAR-10 image classification task, a dramatic improvement over the previous best test accuracy of 40%. Our state-of-the-art results extend across many other settings for MNIST, Fashion-MNIST, CIFAR-10, CIFAR-100, and SVHN. Furthermore, we perform some preliminary analyses of our distilled datasets to shed light on how they differ from naturally occurring data.

cs.LG↗

arXiv2021

On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes

Let $E/\mathbb{Q}$ be an elliptic curve, and $p$ a prime where $E$ has good reduction, and assume that $E$ admits a rational $p$-isogeny. In this paper, we study the anticyclotomic Iwasawa theory of $E$ over an imaginary quadratic field in which $p$ splits, which we relate to the anticyclotomic Iwasawa theory of characters following the method of Greenberg--Vatsal. As a result of our study, we obtain a proof, under mild hypotheses, of Perrin-Riou's Heegner point main conjecture, as well as a $p$-converse to the theorem of Gross--Zagier and Kolyvagin and the $p$-part of the Birch--Swinnerton-Dyer formula in analytic rank $1$ for Eisenstein primes $p$.

math.NT↗

arXiv2021

OCT-GAN: Neural ODE-based Conditional Tabular GANs

Synthesizing tabular data is attracting much attention these days for various purposes. With sophisticate synthetic data, for instance, one can augment its training data. For the past couple of years, tabular data synthesis techniques have been greatly improved. Recent work made progress to address many problems in synthesizing tabular data, such as the imbalanced distribution and multimodality problems. However, the data utility of state-of-the-art methods is not satisfactory yet. In this work, we significantly improve the utility by designing our generator and discriminator based on neural ordinary differential equations (NODEs). After showing that NODEs have theoretically preferred characteristics for generating tabular data, we introduce our designs. The NODE-based discriminator performs a hidden vector evolution trajectory-based classification rather than classifying with a hidden vector at the last layer only. Our generator also adopts an ODE layer at the very beginning of its architecture to transform its initial input vector (i.e., the concatenation of a noisy vector and a condition vector in our case) onto another latent vector space suitable for the generation process. We conduct experiments with 13 datasets, including but not limited to insurance fraud detection, online news article prediction, and so on, and our presented method outperforms other state-of-the-art tabular data synthesis methods in many cases of our classification, regression, and clustering experiments.

cs.LG↗

arXiv2021

A new approach to the Fraser-Li conjecture with the Weierstrass representation formula

In this paper, we provide a sufficient condition for a curve on a surface in $\mathbb{R}^3$ to be given by an orthogonal intersection with a sphere. This result makes it possible to express the boundary condition entirely in terms of the Weierstrass data without integration when dealing with free boundary minimal surfaces in a ball $\mathbb{B}^3$. Moreover, we show that the Gauss map of an embedded free boundary minimal annulus is one to one. By using this, the Fraser-Li conjecture can be translated into the problem of determining the Gauss map. On the other hand, we show that the Liouville type boundary value problem in an annulus gives some new insight into the structure of immersed minimal annuli orthogonal to spheres. It also suggests a new PDE theoretic approach to the Fraser-Li conjecture.

math.DG↗

arXiv2021

Dataset Meta-Learning from Kernel Ridge-Regression

One of the most fundamental aspects of any machine learning algorithm is the training data used by the algorithm. We introduce the novel concept of $ε$-approximation of datasets, obtaining datasets which are much smaller than or are significant corruptions of the original training data while maintaining similar model performance. We introduce a meta-learning algorithm called Kernel Inducing Points (KIP) for obtaining such remarkable datasets, inspired by the recent developments in the correspondence between infinitely-wide neural networks and kernel ridge-regression (KRR). For KRR tasks, we demonstrate that KIP can compress datasets by one or two orders of magnitude, significantly improving previous dataset distillation and subset selection methods while obtaining state of the art results for MNIST and CIFAR-10 classification. Furthermore, our KIP-learned datasets are transferable to the training of finite-width neural networks even beyond the lazy-training regime, which leads to state of the art results for neural network dataset distillation with potential applications to privacy-preservation.

cs.LG↗

arXiv2021

Minimal surfaces in $\mathbb{R}^4$ like the Lagrangian catenoid

In this paper, we discuss complete minimal immersions in $\mathbb{R}^N$($N\geq4$) with finite total curvature and embedded planar ends. First, we prove nonexistence for the following cases: (1) genus 1 with 2 embedded planar ends, (2) genus $\neq4$, hyperelliptic with 2 embedded planar ends like the Lagrangian catenoid. Then we show the existence of embedded minimal spheres in $\mathbb{R}^4$ with 3 embedded planar ends. Moreover, we construct genus $g$ examples in $\mathbb{R}^4$ with $d$ embedded planar ends such that $g\geq 1$ and $g+2\leq d\leq 2g+1$. These examples include a family of embedded minimal tori with 3 embedded planar ends.

math.DG↗

arXiv2020

Towards NNGP-guided Neural Architecture Search

The predictions of wide Bayesian neural networks are described by a Gaussian process, known as the Neural Network Gaussian Process (NNGP). Analytic forms for NNGP kernels are known for many models, but computing the exact kernel for convolutional architectures is prohibitively expensive. One can obtain effective approximations of these kernels through Monte-Carlo estimation using finite networks at initialization. Monte-Carlo NNGP inference is orders-of-magnitude cheaper in FLOPs compared to gradient descent training when the dataset size is small. Since NNGP inference provides a cheap measure of performance of a network architecture, we investigate its potential as a signal for neural architecture search (NAS). We compute the NNGP performance of approximately 423k networks in the NAS-bench 101 dataset on CIFAR-10 and compare its utility against conventional performance measures obtained by shortened gradient-based training. We carry out a similar analysis on 10k randomly sampled networks in the mobile neural architecture search (MNAS) space for ImageNet. We discover comparative advantages of NNGP-based metrics, and discuss potential applications. In particular, we propose that NNGP performance is an inexpensive signal independent of metrics obtained from training that can either be used for reducing big search spaces, or improving training-based performance measures.

cs.LG↗

arXiv2020

Exploring the Uncertainty Properties of Neural Networks' Implicit Priors in the Infinite-Width Limit

Modern deep learning models have achieved great success in predictive accuracy for many data modalities. However, their application to many real-world tasks is restricted by poor uncertainty estimates, such as overconfidence on out-of-distribution (OOD) data and ungraceful failing under distributional shift. Previous benchmarks have found that ensembles of neural networks (NNs) are typically the best calibrated models on OOD data. Inspired by this, we leverage recent theoretical advances that characterize the function-space prior of an ensemble of infinitely-wide NNs as a Gaussian process, termed the neural network Gaussian process (NNGP). We use the NNGP with a softmax link function to build a probabilistic model for multi-class classification and marginalize over the latent Gaussian outputs to sample from the posterior. This gives us a better understanding of the implicit prior NNs place on function space and allows a direct comparison of the calibration of the NNGP and its finite-width analogue. We also examine the calibration of previous approaches to classification with the NNGP, which treat classification problems as regression to the one-hot labels. In this case the Bayesian posterior is exact, and we compare several heuristics to generate a categorical distribution over classes. We find these methods are well calibrated under distributional shift. Finally, we consider an infinite-width final layer in conjunction with a pre-trained embedding. This replicates the important practical use case of transfer learning and allows scaling to significantly larger datasets. As well as achieving competitive predictive accuracy, this approach is better calibrated than its finite width analogue.

stat.ML↗

arXiv2020

Finite Versus Infinite Neural Networks: an Empirical Study

We perform a careful, thorough, and large scale empirical study of the correspondence between wide neural networks and kernel methods. By doing so, we resolve a variety of open questions related to the study of infinitely wide neural networks. Our experimental results include: kernel methods outperform fully-connected finite-width networks, but underperform convolutional finite width networks; neural network Gaussian process (NNGP) kernels frequently outperform neural tangent (NT) kernels; centered and ensembled finite networks have reduced posterior variance and behave more similarly to infinite networks; weight decay and the use of a large learning rate break the correspondence between finite and infinite networks; the NTK parameterization outperforms the standard parameterization for finite width networks; diagonal regularization of kernels acts similarly to early stopping; floating point precision limits kernel performance beyond a critical dataset size; regularized ZCA whitening improves accuracy; finite network performance depends non-monotonically on width in ways not captured by double descent phenomena; equivariance of CNNs is only beneficial for narrow networks far from the kernel regime. Our experiments additionally motivate an improved layer-wise scaling for weight decay which improves generalization in finite-width networks. Finally, we develop improved best practices for using NNGP and NT kernels for prediction, including a novel ensembling technique. Using these best practices we achieve state-of-the-art results on CIFAR-10 classification for kernels corresponding to each architecture class we consider.

cs.LG↗

arXiv2020

Bayesian Deep Convolutional Networks with Many Channels are Gaussian Processes

There is a previously identified equivalence between wide fully connected neural networks (FCNs) and Gaussian processes (GPs). This equivalence enables, for instance, test set predictions that would have resulted from a fully Bayesian, infinitely wide trained FCN to be computed without ever instantiating the FCN, but by instead evaluating the corresponding GP. In this work, we derive an analogous equivalence for multi-layer convolutional neural networks (CNNs) both with and without pooling layers, and achieve state of the art results on CIFAR10 for GPs without trainable kernels. We also introduce a Monte Carlo method to estimate the GP corresponding to a given neural network architecture, even in cases where the analytic form has too many terms to be computationally feasible. Surprisingly, in the absence of pooling layers, the GPs corresponding to CNNs with and without weight sharing are identical. As a consequence, translation equivariance, beneficial in finite channel CNNs trained with stochastic gradient descent (SGD), is guaranteed to play no role in the Bayesian treatment of the infinite channel limit - a qualitative difference between the two regimes that is not present in the FCN case. We confirm experimentally, that while in some scenarios the performance of SGD-trained finite CNNs approaches that of the corresponding GPs as the channel count increases, with careful tuning SGD-trained CNNs can significantly outperform their corresponding GPs, suggesting advantages from SGD training compared to fully Bayesian parameter estimation.

stat.ML↗

arXiv2020

Data-Efficient Deep Learning Method for Image Classification Using Data Augmentation, Focal Cosine Loss, and Ensemble

In general, sufficient data is essential for the better performance and generalization of deep-learning models. However, lots of limitations(cost, resources, etc.) of data collection leads to lack of enough data in most of the areas. In addition, various domains of each data sources and licenses also lead to difficulties in collection of sufficient data. This situation makes us hard to utilize not only the pre-trained model, but also the external knowledge. Therefore, it is important to leverage small dataset effectively for achieving the better performance. We applied some techniques in three aspects: data, loss function, and prediction to enable training from scratch with less data. With these methods, we obtain high accuracy by leveraging ImageNet data which consist of only 50 images per class. Furthermore, our model is ranked 4th in Visual Inductive Printers for Data-Effective Computer Vision Challenge.

cs.CV↗

arXiv2020

Closed Lagrangian self-shrinkers in $\mathbb{R}^4$ symmetric with respect to a hyperplane

In this paper, we prove that the closed Lagrangian self-shrinkers in $\mathbb{R}^4$ which are symmetric with respect to a hyperplane are given by the products of Abresch-Langer curves. As a corollary, we obtain a new geometric characterization of the Clifford torus as the unique embedded closed Lagrangian self-shrinker symmetric with respect to a hyperplane in $\mathbb{R}^4$.

math.DG↗