A Tight Parallel Repetition Theorem for Partially Simulatable Interactive Arguments via Smooth KL-Divergence

Hardness amplification is a central problem in the study of interactive protocols. While natural parallel repetition transformation is known to reduce the soundness error of some special cases of interactive arguments: three-message protocols (Bellare, Impagliazzo, and Naor [FOCS \u2797]) and public-coin protocols (Hastad, Pass, Wikstrom, and Pietrzak [TCC \u2710], Chung and Lu [TCC \u2710] and Chung and Pass [TCC \u2715]), it fails to do so in the general case (the above Bellare et al.; also Pietrzak and Wikstrom [TCC \u2707]). The only known round-preserving approach that applies to all interactive arguments is Haitner\u27s random-terminating transformation [SICOMP \u2713], who showed that the parallel repetition of the transformed protocol reduces the soundness error at a weak exponential rate: if the original $m$-round protocol has soundness error $1-\varepsilon$, then the $n$-parallel repetition of its random-terminating variant has soundness error $(1-\varepsilon)^{\varepsilon n / m^4}$ (omitting constant factors). Hastad et al. have generalized this result to partially simulatable interactive arguments, showing that the $n$-fold repetition of an $m$-round $\delta$-simulatable argument of soundness error $1-\varepsilon$ has soundness error $(1-\varepsilon)^{\varepsilon \delta^2 n / m^2}$. When applied to random-terminating arguments, the Hastad et al. bound matches that of Haitner. In this work we prove that parallel repetition of random-terminating arguments reduces the soundness error at a much stronger exponential rate: the soundness error of the $n$ parallel repetition is $(1-\varepsilon)^{n / m}$, only an $m$ factor from the optimal rate of $(1-\varepsilon)^n$ achievable in public-coin and three-message arguments. The result generalizes to $\delta$-simulatable arguments, for which we prove a bound of $(1-\varepsilon)^{\delta n / m}$. This is achieved by presenting a tight bound on a relaxed variant of the KL-divergence between the distribution induced by our reduction and its ideal variant, a result whose scope extends beyond parallel repetition proofs. We prove the tightness of the above bound for random-terminating arguments, by presenting a matching protocol

A comparison of the CAR and DAGAR spatial random effects models with an application to diabetics rate estimation in Belgium

When hierarchically modelling an epidemiological phenomenon on a finite collection of sites in space, one must always take a latent spatial effect into account in order to capture the correlation structure that links the phenomenon to the territory. In this work, we compare two autoregressive spatial models that can be used for this purpose: the classical CAR model and the more recent DAGAR model. Differently from the former, the latter has a desirable property: its ρ parameter can be naturally interpreted as the average neighbor pair correlation and, in addition, this parameter can be directly estimated when the effect is modelled using a DAGAR rather than a CAR structure. As an application, we model the diabetics rate in Belgium in 2014 and show the adequacy of these models in predicting the response variable when no covariates are available

A Statistical Approach to the Alignment of fMRI Data

Multi-subject functional Magnetic Resonance Image studies are critical. The anatomical and functional structure varies across subjects, so the image alignment is necessary. We define a probabilistic model to describe functional alignment. Imposing a prior distribution, as the matrix Fisher Von Mises distribution, of the orthogonal transformation parameter, the anatomical information is embedded in the estimation of the parameters, i.e., penalizing the combination of spatially distant voxels. Real applications show an improvement in the classification and interpretability of the results compared to various functional alignment methods