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We consider general regularisation and regularised interpolation problems for learning parameter vectors from data. In particular in Hilbert spaces regularisation methods have been applied very successfully, largely due to the well known representer theorem. Classical formulations of the theorem state that under certain conditions on the regulariser there exists a solution of the optimisation problem which is contained in a linear subspace spanned by the data points. This is at the core of kernel methods in machine learning as it significantly reduces the dimensionality and thus makes the problem computationally tractable. Most of the literature only deals with sufficient conditions on the regulariser for a representer theorem to hold, mostly in Hilbert spaces with some generalisations to certain classes of Banach spaces. In this work we give an essentially complete answer to the question of existence of representer theorems in general Banach spaces. This question had previously been answered for Hilbert spaces with an intuitive characterisation for differentiable regularisers. We show how the necessary and sufficient conditions extend to arbitrary Banach spaces and give the more intuitive geometric characterisation for a variety of classes of Banach spaces, which contain all spaces we know of which are commonly used in applications. We conjecture that the same characterisation can also be given for any other Banach space not currently covered by those classes. We further show that, if the learning relies on the linear representer theorem, in most cases the solution is actually independent of the regulariser and determined by the function space alone. This is interesting for two reasons. Firstly it means one is free to choose whichever regulariser is most suitable for the application at hand, whether this is computational efficiency or ease of calculations. Moreover it shows the importance of extending classical elements of learning theory such as kernel methods from Hilbert spaces to Banach spaces.

This paper contributes to the challenge of skeleton-based human action recognition in videos. The key step is to develop a generic network architecture to extract discriminative features for the spatio-temporal skeleton data. In this paper, we propose a novel module, namely Logsig-RNN, which is the combination of the log-signature layer and recurrent type neural networks (RNNs). The former one comes from the mathematically principled technology of signatures and log-signatures as representations for streamed data, which can manage high sample rate streams, non-uniform sampling and time series of variable length. It serves as an enhancement of the recurrent layer, which can be conveniently plugged into neural networks. Besides we propose two path transformation layers to significantly reduce path dimension while retaining the essential information fed into the Logsig-RNN module. Finally, numerical results demonstrate that replacing the RNN module by the Logsig-RNN module in SOTA networks consistently improves the performance on both Chalearn gesture data and NTU RGB+D 120 action data in terms of accuracy and robustness. In particular, we achieve the state-of-the-art accuracy on Chalearn2013 gesture data by combining simple path transformation layers with the Logsig-RNN. Codes are available at https://github.com/steveliao93/GCN_LogsigRNN., This paper is accepted by British Machine Vision Conference 2021

People high in autistic-like traits have been found to have difficulties with recognizing emotions from nonverbal expressions. However, findings on the autism—emotion recognition relationship are inconsistent. In the present study, we investigated whether speeded reasoning ability (reasoning performance under time pressure) moderates the inverse relationship between autistic-like traits and emotion recognition performance. We expected the negative correlation between autistic-like traits and emotion recognition to be less strong when speeded reasoning ability was high. MTurkers (N = 217) completed the ten item version of the Autism Spectrum Quotient (AQ-10), two emotion recognition tests using videos with sound (Geneva Emotion Recognition Test, GERT-S) and pictures (Reading the Mind in the Eyes Test, RMET), and Baddeley's Grammatical Reasoning test to measure speeded reasoning. As expected, the higher the ability in speeded reasoning, the less were higher autistic-like traits related to lower emotion recognition performance. These results suggest that a high ability in making quick mental inferences may (partly) compensate for difficulties with intuitive emotion recognition related to autistic-like traits.