A Neuro-Symbolic Method for Solving Differential and Functional Equations

When neural networks are used to solve differential equations, they usually produce solutions in the form of black-box functions that are not directly mathematically interpretable. We introduce a method for generating symbolic expressions to solve differential equations while leveraging deep learning training methods. Unlike existing methods, our system does not require learning a language model over symbolic mathematics, making it scalable, compact, and easily adaptable for a variety of tasks and configurations. As part of the method, we propose a novel neural architecture for learning mathematical expressions to optimize a customizable objective. The system is designed to always return a valid symbolic formula, generating a useful approximation when an exact analytic solution to a differential equation is not or cannot be found. We demonstrate through examples how our method can be applied on a number of differential equations, often obtaining symbolic approximations that are useful or insightful. Furthermore, we show how the system can be effortlessly generalized to find symbolic solutions to other mathematical tasks, including integration and functional equations.

Stochastic Neighbor Embedding with Gaussian and Student-t Distributions: Tutorial and Survey

Stochastic Neighbor Embedding (SNE) is a manifold learning and dimensionality reduction method with a probabilistic approach. In SNE, every point is consider to be the neighbor of all other points with some probability and this probability is tried to be preserved in the embedding space. SNE considers Gaussian distribution for the probability in both the input and embedding spaces. However, t-SNE uses the Student-t and Gaussian distributions in these spaces, respectively. In this tutorial and survey paper, we explain SNE, symmetric SNE, t-SNE (or Cauchy-SNE), and t-SNE with general degrees of freedom. We also cover the out-of-sample extension and acceleration for these methods. Some simulations to visualize the embeddings are also provided.

Multidimensional Scaling, Sammon Mapping, and Isomap: Tutorial and Survey

Multidimensional Scaling (MDS) is one of the first fundamental manifold learning methods. It can be categorized into several methods, i.e., classical MDS, kernel classical MDS, metric MDS, and non-metric MDS. Sammon mapping and Isomap can be considered as special cases of metric MDS and kernel classical MDS, respectively. In this tutorial and survey paper, we review the theory of MDS, Sammon mapping, and Isomap in detail. We explain all the mentioned categories of MDS. Then, Sammon mapping, Isomap, and kernel Isomap are explained. Out-of-sample embedding for MDS and Isomap using eigenfunctions and kernel mapping are introduced. Then, Nystrom approximation and its use in landmark MDS and landmark Isomap are introduced for big data embedding. We also provide some simulations for illustrating the embedding by these methods.

Segmentation Approach for Coreference Resolution Task

In coreference resolution, it is important to consider all members of a coreference cluster and decide about all of them at once. This technique can help to avoid losing precision and also in finding long-distance relations. The presented paper is a report of an ongoing study on an idea which proposes a new approach for coreference resolution which can resolve all coreference mentions to a given mention in the document in one pass. This has been accomplished by defining an embedding method for the position of all members of a coreference cluster in a document and resolving all of them for a given mention. In the proposed method, the BERT model has been used for encoding the documents and a head network designed to capture the relations between the embedded tokens. These are then converted to the proposed span position embedding matrix which embeds the position of all coreference mentions in the document. We tested this idea on CoNLL 2012 dataset and although the preliminary results from this method do not quite meet the state-of-the-art results, they are promising and they can capture features like long-distance relations better than the other approaches.

DeepNovoV2: Better de novo peptide sequencing with deep learning

Personalized cancer vaccines are envisioned as the next generation rational cancer immunotherapy. The key step in developing personalized therapeutic cancer vaccines is to identify tumor-specific neoantigens that are on the surface of tumor cells. A promising method for this is through de novo peptide sequencing from mass spectrometry data. In this paper we introduce DeepNovoV2, the state-of-the-art model for peptide sequencing. In DeepNovoV2, a spectrum is directly represented as a set of (m/z, intensity) pairs, therefore it does not suffer from the accuracy-speed/memory trade-off problem. The model combines an order invariant network structure (T-Net) and recurrent neural networks and provides a complete end-to-end training and prediction framework to sequence patterns of peptides. Our experiments on a wide variety of data from different species show that DeepNovoV2 outperforms previous state-of-the-art methods, achieving 13.01-23.95\% higher accuracy at the peptide level.

Deep Variational Sufficient Dimensionality Reduction

We consider the problem of sufficient dimensionality reduction (SDR), where the high-dimensional observation is transformed to a low-dimensional sub-space in which the information of the observations regarding the label variable is preserved. We propose DVSDR, a deep variational approach for sufficient dimensionality reduction. The deep structure in our model has a bottleneck that represent the low-dimensional embedding of the data. We explain the SDR problem using graphical models and use the framework of variational autoencoders to maximize the lower bound of the log-likelihood of the joint distribution of the observation and label. We show that such a maximization problem can be interpreted as solving the SDR problem. DVSDR can be easily adopted to semi-supervised learning setting. In our experiment we show that DVSDR performs competitively on classification tasks while being able to generate novel data samples.

SRP: Efficient class-aware embedding learning for large-scale data via supervised random projections

Supervised dimensionality reduction strategies have been of great interest. However, current supervised dimensionality reduction approaches are difficult to scale for situations characterized by large datasets given the high computational complexities associated with such methods. While stochastic approximation strategies have been explored for unsupervised dimensionality reduction to tackle this challenge, such approaches are not well-suited for accelerating computational speed for supervised dimensionality reduction. Motivated to tackle this challenge, in this study we explore a novel direction of directly learning optimal class-aware embeddings in a supervised manner via the notion of supervised random projections (SRP). The key idea behind SRP is that, rather than performing spectral decomposition (or approximations thereof) which are computationally prohibitive for large-scale data, we instead perform a direct decomposition by leveraging kernel approximation theory and the symmetry of the Hilbert-Schmidt Independence Criterion (HSIC) measure of dependence between the embedded data and the labels. Experimental results on five different synthetic and real-world datasets demonstrate that the proposed SRP strategy for class-aware embedding learning can be very promising in producing embeddings that are highly competitive with existing supervised dimensionality reduction methods (e.g., SPCA and KSPCA) while achieving 1-2 orders of magnitude better computational performance. As such, such an efficient approach to learning embeddings for dimensionality reduction can be a powerful tool for large-scale data analysis and visualization.

Text Classification based on Multiple Block Convolutional Highways

In the Text Classification areas of Sentiment Analysis, Subjectivity/Objectivity Analysis, and Opinion Polarity, Convolutional Neural Networks have gained special attention because of their performance and accuracy. In this work, we applied recent advances in CNNs and propose a novel architecture, Multiple Block Convolutional Highways (MBCH), which achieves improved accuracy on multiple popular benchmark datasets, compared to previous architectures. The MBCH is based on new techniques and architectures including highway networks, DenseNet, batch normalization and bottleneck layers. In addition, to cope with the limitations of existing pre-trained word vectors which are used as inputs for the CNN, we propose a novel method, Improved Word Vectors (IWV). The IWV improves the accuracy of CNNs which are used for text classification tasks.

Robust Locally-Linear Controllable Embedding

Embed-to-control (E2C) is a model for solving high-dimensional optimal control problems by combining variational auto-encoders with locally-optimal controllers. However, the E2C model suffers from two major drawbacks: 1) its objective function does not correspond to the likelihood of the data sequence and 2) the variational encoder used for embedding typically has large variational approximation error, especially when there is noise in the system dynamics. In this paper, we present a new model for learning robust locally-linear controllable embedding (RCE). Our model directly estimates the predictive conditional density of the future observation given the current one, while introducing the bottleneck between the current and future observations. Although the bottleneck provides a natural embedding candidate for control, our RCE model introduces additional specific structures in the generative graphical model so that the model dynamics can be robustly linearized. We also propose a principled variational approximation of the embedding posterior that takes the future observation into account, and thus, makes the variational approximation more robust against the noise. Experimental results show that RCE outperforms the E2C model, and does so significantly when the underlying dynamics is noisy.