A normalizing flow models a complex probability density as an invertible transformation of a simple base density. Flows based on either coupling or autoregressive transforms both offer exact density evaluation and sampling, but rely on the parameterization of an easily invertible elementwise transformation, whose choice determines the flexibility of these models. Building upon recent work, we propose a fully-differentiable module based on monotonic rational-quadratic splines, which enhances the flexibility of both coupling and autoregressive transforms while retaining analytic invertibility. We demonstrate that neural spline flows improve density estimation, variational inference, and generative modeling of images.
A normalizing flow models a complex probability density as an invertible transformation of a simple density. The invertibility means that we can evaluate densities and generate samples from a flow. In practice, autoregressive flow-based models are slow to invert, making either density estimation or sample generation slow. Flows based on coupling transforms are fast for both tasks, but have previously performed less well at density estimation than autoregressive flows. We stack a new coupling transform, based on monotonic cubic splines, with LU-decomposed linear layers. The resulting cubic-spli…
SplineFlow: Flow Matching for Dynamical Systems with B-Spline Interpolants
CISPA Helmholtz Center for Information Security（CISPA海德堡信息安全研究中心）；Department of Computer Science and Technology（计算机科学与技术系）；University of Cambridge（剑桥大学）
Gaussian processes are the leading class of distributions on random functions, but they suffer from well known issues including difficulty scaling and inflexibility with respect to certain shape constraints (such as nonnegativity). Here we propose Deep Random Splines, a flexible class of random functions obtained by transforming Gaussian noise through a deep neural network whose output are the parameters of a spline. Unlike Gaussian processes, Deep Random Splines allow us to readily enforce shape constraints while inheriting the richness and tractability of deep generative models. We also pres…
Despite recent advances in generative modeling for text-to-speech synthesis, these models do not yet have the same fine-grained adjustability of pitch-conditioned deterministic models such as FastPitch and FastSpeech2. Pitch information is not only low-dimensional, but also discontinuous, making it particularly difficult to model in a generative setting. Our work explores several techniques for handling the aforementioned issues in the context of Normalizing Flow models. We also find this problem to be very well suited for Neural Spline flows, which is a highly expressive alternative to the mo…
Normalizing flows attempt to model an arbitrary probability distribution through a set of invertible mappings. These transformations are required to achieve a tractable Jacobian determinant that can be used in high-dimensional scenarios. The first normalizing flow designs used coupling layer mappings built upon affine transformations. The significant advantage of such models is their easy-to-compute inverse. Nevertheless, making use of affine transformations may limit the expressiveness of such models. Recently, invertible piecewise polynomial functions as a replacement for affine transformati…
TabResFlow: A Normalizing Spline Flow Model for Probabilistic Univariate Tabular Regression
Information Systems and Machine Learning Lab (ISMLL), University of Hildesheim（信息系统与机器学习实验室（ISMLL）、希尔德斯海姆大学）；VWFS Data Analytics Research Center（VWFS数据分析研究中心）
University of California Santa Cruz（加州大学圣克鲁兹分校）；Kavli IPMU, the University of Tokyo（东京大学卡弗里数物连携宇宙研究机构）；University of Oxford（牛津大学）
Measurement of the red damping wing of neutral hydrogen in quasar spectra provides a probe of the epoch of reionization in the early Universe. Such quantification requires precise and unbiased estimates of the intrinsic continua near Lyman-$α$ (Ly$α$), a challenging task given the highly variable Ly$α$ emission profiles of quasars. Here, we introduce a fully probabilistic approach to intrinsic continua prediction. We frame the problem as a conditional density estimation task and explicitly model the distribution over plausible blue-side continua ($1190\ \unicode{xC5} \leq λ_{\text{rest}} < 129…
Time series, as frequently the case in neuroscience, are rarely stationary, but often exhibit abrupt changes due to attractor transitions or bifurcations in the dynamical systems producing them. A plethora of methods for detecting such change points in time series statistics have been developed over the years, in addition to test criteria to evaluate their significance. Issues to consider when developing change point analysis methods include computational demands, difficulties arising from either limited amount of data or a large number of covariates, and arriving at statistical tests with suf…
Distance-Aware Error for Spline Networks: A Bottom-Up Approach to Uncertainty
Department of Electrical and Computer Engineering of University of Maine（缅因大学电气与计算机工程系）；Electrical and Microelectronic Engineering Department of Rochester Institute of Technology（罗切斯特理工学院电气与微电子工程系）
Waveflow: boundary-conditioned normalizing flows applied to fermionic wavefunctions
University of Toronto（多伦多大学）；Vector Institute for Artificial Intelligence（向量人工智能研究所）；Canadian Institute for Advanced Research (CIFAR)（加拿大高等研究院）；Acceleration Consortium（加速联盟）
We develop a variational framework to understand the properties of functions learned by fitting deep neural networks with rectified linear unit activations to data. We propose a new function space, which is reminiscent of classical bounded variation-type spaces, that captures the compositional structure associated with deep neural networks. We derive a representer theorem showing that deep ReLU networks are solutions to regularized data fitting problems over functions from this space. The function space consists of compositions of functions from the Banach spaces of second-order bounded variat…
We develop a variational framework to understand the properties of the functions learned by neural networks fit to data. We propose and study a family of continuous-domain linear inverse problems with total variation-like regularization in the Radon domain subject to data fitting constraints. We derive a representer theorem showing that finite-width, single-hidden layer neural networks are solutions to these inverse problems. We draw on many techniques from variational spline theory and so we propose the notion of polynomial ridge splines, which correspond to single-hidden layer neural network…
Baylor College of Medicine（贝勒医学院）；Rice University（莱斯大学）
Understanding the learning dynamics and inductive bias of neural networks (NNs) is hindered by the opacity of the relationship between NN parameters and the function represented. We propose reparametrizing ReLU NNs as continuous piecewise linear splines. Using this spline lens, we study learning dynamics in shallow univariate ReLU NNs, finding unexpected insights and explanations for several perplexing phenomena. We develop a surprisingly simple and transparent view of the structure of the loss surface, including its critical and fixed points, Hessian, and Hessian spectrum. We also show that s…
University of Rochester（罗切斯特大学）；Stony Brook University（石溪大学）
The marriage of recurrent neural networks and neural ordinary differential networks (ODE-RNN) is effective in modeling irregularly-observed sequences. While ODE produces the smooth hidden states between observation intervals, the RNN will trigger a hidden state jump when a new observation arrives, thus cause the interpolation discontinuity problem. To address this issue, we propose the cubic spline smoothing compensation, which is a stand-alone module upon either the output or the hidden state of ODE-RNN and can be trained end-to-end. We derive its analytical solution and provide its theoretic…
Inria（法国国家信息与自动化研究所）；Ecole Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）
We consider multivariate splines and show that they have a random feature expansion as infinitely wide neural networks with one-hidden layer and a homogeneous activation function which is the power of the rectified linear unit. We show that the associated function space is a Sobolev space on a Euclidean ball, with an explicit bound on the norms of derivatives. This link provides a new random feature expansion for multivariate splines that allow efficient algorithms. This random feature expansion is numerically better behaved than usual random Fourier features, both in theory and practice. In p…
Australian National University（澳大利亚国立大学）；Data61, CSIRO（联邦科学与工业研究组织 Data61）；Google Research（谷歌研究院）
Calibrating neural networks is of utmost importance when employing them in safety-critical applications where the downstream decision making depends on the predicted probabilities. Measuring calibration error amounts to comparing two empirical distributions. In this work, we introduce a binning-free calibration measure inspired by the classical Kolmogorov-Smirnov (KS) statistical test in which the main idea is to compare the respective cumulative probability distributions. From this, by approximating the empirical cumulative distribution using a differentiable function via splines, we obtain a…
A survey of probabilistic generative frameworks for molecular simulations
University of Maryland（马里兰大学）；Institute for Physical Science and Technology（物理科学与技术研究所）；University of Maryland Institute for Health Computing（马里兰大学健康计算研究所）；Department of Chemistry and Biochemistry（化学与生物化学系）
Deep neural networks (DNNs) generate much richer function spaces than shallow networks. Since the function spaces induced by shallow networks have several approximation theoretic drawbacks, this explains, however, not necessarily the success of deep networks. In this article we take another route by comparing the expressive power of DNNs with ReLU activation function to piecewise linear spline methods. We show that MARS (multivariate adaptive regression splines) is improper learnable by DNNs in the sense that for any given function that can be expressed as a function in MARS with $M$ parameter…
École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
We propose to optimize the activation functions of a deep neural network by adding a corresponding functional regularization to the cost function. We justify the use of a second-order total-variation criterion. This allows us to derive a general representer theorem for deep neural networks that makes a direct connection with splines and sparsity. Specifically, we show that the optimal network configuration can be achieved with activation functions that are nonuniform linear splines with adaptive knots. The bottom line is that the action of each neuron is encoded by a spline whose parameters (i…
A wide variety of activation functions have been proposed for neural networks. The Rectified Linear Unit (ReLU) is especially popular today. There are many practical reasons that motivate the use of the ReLU. This paper provides new theoretical characterizations that support the use of the ReLU, its variants such as the leaky ReLU, as well as other activation functions in the case of univariate, single-hidden layer feedforward neural networks. Our results also explain the importance of commonly used strategies in the design and training of neural networks such as "weight decay" and "path-norm"…
University of Basel（巴塞尔大学）；University of Basel Hospital（巴塞尔大学医院）；Inselspital, Bern University Hospital（伯尔尼大学医院 Inselspital）；University of Bern（伯尔尼大学）
Parametric spatial transformation models have been successfully applied to image registration tasks. In such models, the transformation of interest is parameterized by a fixed set of basis functions as for example B-splines. Each basis function is located on a fixed regular grid position among the image domain, because the transformation of interest is not known in advance. As a consequence, not all basis functions will necessarily contribute to the final transformation which results in a non-compact representation of the transformation. We reformulate the pairwise registration problem as a re…
