We introduce the concept of scalable neural network kernels (SNNKs), the replacements of regular feedforward layers (FFLs), capable of approximating the latter, but with favorable computational properties. SNNKs effectively disentangle the inputs from the parameters of the neural network in the FFL, only to connect them in the final computation via the dot-product kernel. They are also strictly more expressive, as allowing to model complicated relationships beyond the functions of the dot-products of parameter-input vectors. We also introduce the neural network bundling process that applies SNNKs to compactify deep neural network architectures, resulting in additional compression gains. In its extreme version, it leads to the fully bundled network whose optimal parameters can be expressed via explicit formulae for several loss functions (e.g. mean squared error), opening a possibility to bypass backpropagation. As a by-product of our analysis, we introduce the mechanism of the universal random features (or URFs), applied to instantiate several SNNK variants, and interesting on its own in the context of scalable kernel methods. We provide rigorous theoretical analysis of all these concepts as well as an extensive empirical evaluation, ranging from point-wise kernel estimation to Transformers' fine-tuning with novel adapter layers inspired by SNNKs. Our mechanism provides up to 5x reduction in the number of trainable parameters, while maintaining competitive accuracy.
We present two new classes of algorithms for efficient field integration on graphs encoding point clouds. The first class, SeparatorFactorization(SF), leverages the bounded genus of point cloud mesh graphs, while the second class, RFDiffusion(RFD), uses popular epsilon-nearest-neighbor graph representations for point clouds. Both can be viewed as providing the functionality of Fast Multipole Methods (FMMs), which have had a tremendous impact on efficient integration, but for non-Euclidean spaces. We focus on geometries induced by distributions of walk lengths between points (e.g., shortest-path distance). We provide an extensive theoretical analysis of our algorithms, obtaining new results in structural graph theory as a byproduct. We also perform exhaustive empirical evaluation, including on-surface interpolation for rigid and deformable objects (particularly for mesh-dynamics modeling), Wasserstein distance computations for point clouds, and the Gromov-Wasserstein variant.
Pretrained Transformer based models finetuned on domain specific corpora have changed the landscape of NLP. However, training or fine-tuning these models for individual tasks can be time consuming and resource intensive. Thus, a lot of current research is focused on using transformers for multi-task learning (Raffel et al.,2020) and how to group the tasks to help a multi-task model to learn effective representations that can be shared across tasks (Standley et al., 2020; Fifty et al., 2021). In this work, we show that a single multi-tasking model can match the performance of task specific models when the task specific models show similar representations across all of their hidden layers and their gradients are aligned, i.e. their gradients follow the same direction. We hypothesize that the above observations explain the effectiveness of multi-task learning. We validate our observations on our internal radiologist-annotated datasets on the cervical and lumbar spine. Our method is simple and intuitive, and can be used in a wide range of NLP problems.
Pretrained Transformer based models finetuned on domain specific corpora have changed the landscape of NLP. Generally, if one has multiple tasks on a given dataset, one may finetune different models or use task specific adapters. In this work, we show that a multi-task model can beat or achieve the performance of multiple BERT-based models finetuned on various tasks and various task specific adapter augmented BERT-based models. We validate our method on our internal radiologist's report dataset on cervical spine. We hypothesize that the tasks are semantically close and related and thus multitask learners are powerful classifiers. Our work opens the scope of using our method to radiologist's reports on various body parts.
We propose a new class of random feature methods for linearizing softmax and Gaussian kernels called hybrid random features (HRFs) that automatically adapt the quality of kernel estimation to provide most accurate approximation in the defined regions of interest. Special instantiations of HRFs lead to well-known methods such as trigonometric (Rahimi and Recht, 2007) or (recently introduced in the context of linear-attention Transformers) positive random features (Choromanski et al., 2021). By generalizing Bochner's Theorem for softmax/Gaussian kernels and leveraging random features for compositional kernels, the HRF-mechanism provides strong theoretical guarantees - unbiased approximation and strictly smaller worst-case relative errors than its counterparts. We conduct exhaustive empirical evaluation of HRF ranging from pointwise kernel estimation experiments, through tests on data admitting clustering structure to benchmarking implicit-attention Transformers (also for downstream Robotics applications), demonstrating its quality in a wide spectrum of machine learning problems.
Transformers are state-of-the-art deep learning models that are composed of stacked attention and point-wise, fully connected layers designed for handling sequential data. Transformers are not only ubiquitous throughout Natural Language Processing (NLP), but, recently, they have inspired a new wave of Computer Vision (CV) applications research. In this work, a Vision Transformer (ViT) is applied to predict the state variables of 2-dimensional Ising model simulations. Our experiments show that ViT outperform state-of-the-art Convolutional Neural Networks (CNN) when using a small number of microstate images from the Ising model corresponding to various boundary conditions and temperatures. This work opens the possibility of applying ViT to other simulations, and raises interesting research directions on how attention maps can learn about the underlying physics governing different phenomena.
In this report, the application of the Quantum Potential Neural Network (QPNN) framework to many electron atomic systems is presented. For this study, full configuration interaction (FCI) one--electron density functions within predefined limits of accuracy were used to train the QPNN. The obtained results suggest that this new neural network is capable of learning the effective potential functions of many electron atoms in a completely unsupervised manner, and using only limited information from the probability density. Using the effective potential functions learned for each of the studied systems the QPNN was able to estimate the total energies of each of the systems (with a maximum of 10 trials) with a remarkable accuracy when compared to the FCI energies.
Understanding the space of probability measures on a metric space equipped with a Wasserstein distance is one of the fundamental questions in mathematical analysis. The Wasserstein metric has received a lot of attention in the machine learning community especially for its principled way of comparing distributions. In this work, we use a permutation invariant network to map samples from probability measures into a low-dimensional space such that the Euclidean distance between the encoded samples reflects the Wasserstein distance between probability measures. We show that our network can generalize to correctly compute distances between unseen densities. We also show that these networks can learn the first and the second moments of probability distributions.