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Register a new usergiotto-tda: A Topological Data Analysis Toolkit for Machine Learning and Data Exploration
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Added by
Guillaume Tauzin
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We introduce giotto-tda, a Python library that integrates high-performance topological data analysis with machine learning via a scikit-learn-compatible API and state-of-the-art C++ implementations. The librarys ability to handle various types of data is rooted in a wide range of preprocessing techniques, and its strong focus on data exploration and interpretability is aided by an intuitive plotting API. Source code, binaries, examples, and documentation can be found at https://github.com/giotto-ai/giotto-tda.
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Statistical analysis on object data presents many challenges. Basic summaries such as means and variances are difficult to compute. We apply ideas from topology to study object data. We present a framework for using persistence landscapes to vectorize object data and perform statistical analysis. We apply to this pipeline to some biological images that were previously shown to be challenging to study using shape theory. Surprisingly, the most persistent features are shown to be topological noise and the statistical analysis depends on the less persistent features which we refer to as the geometric signal. We also describe the first steps to a new approach to using topology for object data analysis, which applies topology to distributions on object spaces.
A relatively new set of transport-based transforms (CDT, R-CDT, LOT) have shown their strength and great potential in various image and data processing tasks such as parametric signal estimation, classification, cancer detection among many others. It is hence worthwhile to elucidate some of the mathematical properties that explain the successes of these transforms when they are used as tools in data analysis, signal processing or data classification. In particular, we give conditions under which classes of signals that are created by algebraic generative models are transformed into convex sets by the transport transforms. Such convexification of the classes simplify the classification and other data analysis and processing problems when viewed in the transform domain. More specifically, we study the extent and limitation of the convexification ability of these transforms under an algebraic generative modeling framework. We hope that this paper will serve as an introduction to these transforms and will encourage mathematicians and other researchers to further explore the theoretical underpinnings and algorithmic tools that will help understand the successes of these transforms and lay the groundwork for further successful applications.
Topological data analysis (TDA) has emerged as one of the most promising techniques to reconstruct the unknown shapes of high-dimensional spaces from observed data samples. TDA, thus, yields key shape descriptors in the form of persistent topological features that can be used for any supervised or unsupervised learning task, including multi-way classification. Sparse sampling, on the other hand, provides a highly efficient technique to reconstruct signals in the spatial-temporal domain from just a few carefully-chosen samples. Here, we present a new method, referred to as the Sparse-TDA algorithm, that combines favorable aspects of the two techniques. This combination is realized by selecting an optimal set of sparse pixel samples from the persistent features generated by a vector-based TDA algorithm. These sparse samples are selected from a low-rank matrix representation of persistent features using QR pivoting. We show that the Sparse-TDA method demonstrates promising performance on three benchmark problems related to human posture recognition and image texture classification.
The usability and practicality of any machine learning (ML) applications are largely influenced by two critical but hard-to-attain factors: low latency and low cost. Unfortunately, achieving low latency and low cost is very challenging when ML depends on real-world data that are highly distributed and rapidly growing (e.g., data collected by mobile phones and video cameras all over the world). Such real-world data pose many challenges in communication and computation. For example, when training data are distributed across data centers that span multiple continents, communication among data centers can easily overwhelm the limited wide-area network bandwidth, leading to prohibitively high latency and high cost. In this dissertation, we demonstrate that the latency and cost of ML on highly-distributed and rapidly-growing data can be improved by one to two orders of magnitude by designing ML systems that exploit the characteristics of ML algorithms, ML model structures, and ML training/serving data. We support this thesis statement with three contributions. First, we design a system that provides both low-latency and low-cost ML serving (inferencing) over large-scale and continuously-growing datasets, such as videos. Second, we build a system that makes ML training over geo-distributed datasets as fast as training within a single data center. Third, we present a first detailed study and a system-level solution on a fundamental and largely overlooked problem: ML training over non-IID (i.e., not independent and identically distributed) data partitions (e.g., facial images collected by cameras varies according to the demographics of each cameras location).
Incremental gradient (IG) methods, such as stochastic gradient descent and its variants are commonly used for large scale optimization in machine learning. Despite the sustained effort to make IG methods more data-efficient, it remains an open question how to select a training data subset that can theoretically and practically perform on par with the full dataset. Here we develop CRAIG, a method to select a weighted subset (or coreset) of training data that closely estimates the full gradient by maximizing a submodular function. We prove that applying IG to this subset is guaranteed to converge to the (near)optimal solution with the same convergence rate as that of IG for convex optimization. As a result, CRAIG achieves a speedup that is inversely proportional to the size of the subset. To our knowledge, this is the first rigorous method for data-efficient training of general machine learning models. Our extensive set of experiments show that CRAIG, while achieving practically the same solution, speeds up various IG methods by up to 6x for logistic regression and 3x for training deep neural networks.
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