arXiv subjects
Publications and source records attributed to Tom Rush.
For compactly supported $\mu \in \mathcal{P}(\mathrm{SL}(2,\mathbb{R}))$ satisfying strong irreducibility and proximality, we obtain a formula for the Frostman dimension of the associated Furstenberg measure. We also describe the left neighbourhood of 0 for which the classical transfer operators defined by Le Page have a spectral gap on H\"older spaces in this setting.
Let $(\Sigma_T,\sigma)$ be a subshift of finite type with primitive adjacency matrix $T$, $\psi:\Sigma_T \rightarrow \mathbb{R}$ a H\"older continuous potential, and $\mathcal{A}:\Sigma_T \rightarrow \mathrm{GL}_d(\mathbb{R})$ a 1-typical, one-step cocycle. For $t \in \mathbb{R}$ consider the sequences of potentials $\Phi_t=(\varphi_{t,n})_{n \in \mathbb{N}}$ defined by $$\varphi_{t,n}(x):=S_n \psi(x) + t\log \|\mathcal{A}^n(x)\|, \: \forall n \in \mathbb{N}.$$ Using the family of transfer operators defined in this setting by Park and Piraino, for all $t<0$ sufficiently close to 0 we prove the existence of Gibbs-type measures for the superadditive sequences of potentials $\Phi_t$. This extends the results of the well-understood subadditive case where $t \geq 0$. Prior to this, Gibbs-type measures were only known to exist for $t<0$ in the conformal, the reducible, the positive, or the dominated, planar settings, in which case they are Gibbs measures in the classical sense. We further prove that the topological pressure function $t \mapsto P_{\mathrm{top}}(\Phi_t,\sigma)$ is analytic in an open neighbourhood of 0 and has derivative given by the Lyapunov exponents of these Gibbs-type measures.
We study multifractal decompositions based on Birkhoff averages for sequences of functions belonging to certain classes of symbolically continuous functions. We do this for an expanding interval map with countably many branches, which we assume can be coded by a topologically mixing countable Markov shift. This generalises previous work on expanding maps with finitely many branches, and expanding maps with countably many branches where the coding is assumed to be the full shift. When the infimum of the derivative on each branch approaches infinity in the limit, we can directly generalise the results of the full countable shift case. However, when this does not hold, we show that there can be different behaviour, in particular in cases where the coding has finite topological entropy.
One of the key requirements for incorporating machine learning into the drug discovery process is complete reproducibility and traceability of the model building and evaluation process. With this in mind, we have developed an end-to-end modular and extensible software pipeline for building and sharing machine learning models that predict key pharma-relevant parameters. The ATOM Modeling PipeLine, or AMPL, extends the functionality of the open source library DeepChem and supports an array of machine learning and molecular featurization tools. We have benchmarked AMPL on a large collection of pharmaceutical datasets covering a wide range of parameters. As a result of these comprehensive experiments, we have found that physicochemical descriptors and deep learning-based graph representations significantly outperform traditional fingerprints in the characterization of molecular features. We have also found that dataset size is directly correlated to prediction performance, and that single-task deep learning models only outperform shallow learners if there is sufficient data. Likewise, dataset size has a direct impact on model predictivity, independent of comprehensive hyperparameter model tuning. Our findings point to the need for public dataset integration or multi-task/transfer learning approaches. Lastly, we found that uncertainty quantification (UQ) analysis may help identify model error; however, efficacy of UQ to filter predictions varies considerably between datasets and featurization/model types. AMPL is open source and available for download at http://github.com/ATOMconsortium/AMPL.