Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGaussian Processes indexed on the symmetric group: prediction and learning
212
0
0.0
(
0
)
Added by
Baptiste Broto
Publication date
2018
fields
Mathematical Statistics
and research's language is
English
Authors
Franc{c}ois Bachoc
Ask ChatGPT about the research
No Arabic abstract
In the framework of the supervised learning of a real function defined on a space X , the so called Kriging method stands on a real Gaussian field defined on X. The Euclidean case is well known and has been widely studied. In this paper, we explore the less classical case where X is the non commutative finite group of permutations. In this setting, we propose and study an harmonic analysis of the covariance operators that enables to consider Gaussian processes models and forecasting issues. Our theory is motivated by statistical ranking problems.
rate research
Read More
Propagating input uncertainty through non-linear Gaussian process (GP) mappings is intractable. This hinders the task of training GPs using uncertain and partially observed inputs. In this paper we refer to this task as semi-described learning. We then introduce a GP framework that solves both, the semi-described and the semi-supervised learning problems (where missing values occur in the outputs). Auto-regressive state space simulation is also recognised as a special case of semi-described learning. To achieve our goal we develop variational methods for handling semi-described inputs in GPs, and couple them with algorithms that allow for imputing the missing values while treating the uncertainty in a principled, Bayesian manner. Extensive experiments on simulated and real-world data study the problems of iterative forecasting and regression/classification with missing values. The results suggest that the principled propagation of uncertainty stemming from our framework can significantly improve performance in these tasks.
When fitting Bayesian machine learning models on scarce data, the main challenge is to obtain suitable prior knowledge and encode it into the model. Recent advances in meta-learning offer powerful methods for extracting such prior knowledge from data acquired in related tasks. When it comes to meta-learning in Gaussian process models, approaches in this setting have mostly focused on learning the kernel function of the prior, but not on learning its mean function. In this work, we explore meta-learning the mean function of a Gaussian process prior. We present analytical and empirical evidence that mean function learning can be useful in the meta-learning setting, discuss the risk of overfitting, and draw connections to other meta-learning approaches, such as model agnostic meta-learning and functional PCA.
Kernel methods on discrete domains have shown great promise for many challenging data types, for instance, biological sequence data and molecular structure data. Scalable kernel methods like Support Vector Machines may offer good predictive performances but do not intrinsically provide uncertainty estimates. In contrast, probabilistic kernel methods like Gaussian Processes offer uncertainty estimates in addition to good predictive performance but fall short in terms of scalability. While the scalability of Gaussian processes can be improved using sparse inducing point approximations, the selection of these inducing points remains challenging. We explore different techniques for selecting inducing points on discrete domains, including greedy selection, determinantal point processes, and simulated annealing. We find that simulated annealing, which can select inducing points that are not in the training set, can perform competitively with support vector machines and full Gaussian processes on synthetic data, as well as on challenging real-world DNA sequence data.
We introduce a framework for Continual Learning (CL) based on Bayesian inference over the function space rather than the parameters of a deep neural network. This method, referred to as functional regularisation for Continual Learning, avoids forgetting a previous task by constructing and memorising an approximate posterior belief over the underlying task-specific function. To achieve this we rely on a Gaussian process obtained by treating the weights of the last layer of a neural network as random and Gaussian distributed. Then, the training algorithm sequentially encounters tasks and constructs posterior beliefs over the task-specific functions by using inducing point sparse Gaussian process methods. At each step a new task is first learnt and then a summary is constructed consisting of (i) inducing inputs -- a fixed-size subset of the task inputs selected such that it optimally represents the task -- and (ii) a posterior distribution over the function values at these inputs. This summary then regularises learning of future tasks, through Kullback-Leibler regularisation terms. Our method thus unites approaches focused on (pseudo-)rehearsal with those derived from a sequential Bayesian inference perspective in a principled way, leading to strong results on accepted benchmarks.
We present a multi-task learning formulation for Deep Gaussian processes (DGPs), through non-linear mixtures of latent processes. The latent space is composed of private processes that capture within-task information and shared processes that capture across-task dependencies. We propose two different methods for segmenting the latent space: through hard coding shared and task-specific processes or through soft sharing with Automatic Relevance Determination kernels. We show that our formulation is able to improve the learning performance and transfer information between the tasks, outperforming other probabilistic multi-task learning models across real-world and benchmarking settings.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
