Non-asymptotic Optimal Prediction Error for RKHS-based Partially Functional Linear Models

Under the framework of reproducing kernel Hilbert space (RKHS), we consider the penalized least-squares of the partially functional linear models (PFLM), whose predictor contains both functional and traditional multivariate part, and the multivariate part allows a divergent number of parameters. From the non-asymptotic point of view, we focus on the rate-optimal upper and lower bounds of the prediction error. An exact upper bound for the excess prediction risk is shown in a non-asymptotic form under a more general assumption known as the effective dimension to the model, by which we also show the prediction consistency when the number of multivariate covariates $p$ slightly increases with the sample size $n$. Our new finding implies a trade-off between the number of non-functional predictors and the effective dimension of the kernel principal components to ensure the prediction consistency in the increasing-dimensional setting. The analysis in our proof hinges on the spectral condition of the sandwich operator of the covariance operator and the reproducing kernel, and on the concentration inequalities for the random elements in Hilbert space. Finally, we derive the non-asymptotic minimax lower bound under the regularity assumption of Kullback-Leibler divergence of the models.Comment: 24 page

Local Application of Sodium Salicylate Enhances Auditory Responses in the Rat’s Dorsal Cortex of the Inferior Colliculus

Sodium salicylate (SS) is a widely used medication with side effects on hearing. In order to understand these side effects, we recorded sound-driven local-field potentials in a neural structure, the dorsal cortex of the inferior colliculus (ICd). Using a microiontophoretic technique, we applied SS at sites of recording and studied how auditory responses were affected by the drug. Furthermore, we studied how the responses were affected by combined local application of SS and an agonists/antagonist of the type-A or type-B γ-aminobutyric acid receptor (GABAA or GABAB receptor). Results revealed that SS applied alone enhanced auditory responses in the ICd, indicating that the drug had local targets in the structure. Simultaneous application of the drug and a GABAergic receptor antagonist synergistically enhanced amplitudes of responses. The synergistic interaction between SS and a GABAA receptor antagonist had a relatively early start in reference to the onset of acoustic stimulation and the duration of this interaction was independent of sound intensity. The interaction between SS and a GABAB receptor antagonist had a relatively late start, and the duration of this interaction was dependent on sound intensity. Simultaneous application of the drug and a GABAergic receptor agonist produced an effect different from the sum of effects produced by the two drugs released individually. These differences between simultaneous and individual drug applications suggest that SS modified GABAergic inhibition in the ICd. Our results indicate that SS can affect sound-driven activity in the ICd by modulating local GABAergic inhibition