Multiscale relevance of natural images.

We use an agnostic information-theoretic approach to investigate the statistical properties of natural images. We introduce the Multiscale Relevance (MSR) measure to assess the robustness of images to compression at all scales. Starting in a controlled environment, we characterize the MSR of synthetic random textures as function of image roughness [Formula: see text] and other relevant parameters. We then extend the analysis to natural images and find striking similarities with critical ([Formula: see text]) random textures. We show that the MSR is more robust and informative of image content than classical methods such as power spectrum analysis. Finally, we confront the MSR to classical measures for the calibration of common procedures such as color mapping and denoising. Overall, the MSR approach appears to be a good candidate for advanced image analysis and image processing, while providing a good level of physical interpretability.

The Stochastic Complexity of Spin Models: Are Pairwise Models Really Simple?

Models can be simple for different reasons: because they yield a simple and computationally efficient interpretation of a generic dataset (e.g., in terms of pairwise dependencies)-as in statistical learning-or because they capture the laws of a specific phenomenon-as e.g., in physics-leading to non-trivial falsifiable predictions. In information theory, the simplicity of a model is quantified by the stochastic complexity, which measures the number of bits needed to encode its parameters. In order to understand how simple models look like, we study the stochastic complexity of spin models with interactions of arbitrary order. We show that bijections within the space of possible interactions preserve the stochastic complexity, which allows to partition the space of all models into equivalence classes. We thus found that the simplicity of a model is not determined by the order of the interactions, but rather by their mutual arrangements. Models where statistical dependencies are localized on non-overlapping groups of few variables are simple, affording predictions on independencies that are easy to falsify. On the contrary, fully connected pairwise models, which are often used in statistical learning, appear to be highly complex, because of their extended set of interactions, and they are hard to falsify.

Linear processes in high dimensions: Phase space and critical properties.

In this work we investigate the generic properties of a stochastic linear model in the regime of high dimensionality. We consider in particular the vector autoregressive (VAR) model and the multivariate Hawkes process. We analyze both deterministic and random versions of these models, showing the existence of a stable phase and an unstable phase. We find that along the transition region separating the two regimes the correlations of the process decay slowly, and we characterize the conditions under which these slow correlations are expected to become power laws. We check our findings with numerical simulations showing remarkable agreement with our predictions. We finally argue that real systems with a strong degree of self-interaction are naturally characterized by this type of slow relaxation of the correlations.

Agent-based models for latent liquidity and concave price impact.

We revisit the "ɛ-intelligence" model of Tóth et al. [Phys. Rev. X 1, 021006 (2011)], which was proposed as a minimal framework to understand the square-root dependence of the impact of meta-orders on volume in financial markets. The basic idea is that most of the daily liquidity is "latent" and furthermore vanishes linearly around the current price, as a consequence of the diffusion of the price itself. However, the numerical implementation of Tóth et al. (2011) was criticized as being unrealistic, in particular because all the "intelligence" was conferred to market orders, while limit orders were passive and random. In this work, we study various alternative specifications of the model, for example, allowing limit orders to react to the order flow or changing the execution protocols. By and large, our study lends strong support to the idea that the square-root impact law is a very generic and robust property that requires very few ingredients to be valid. We also show that the transition from superdiffusion to subdiffusion reported in Tóth et al. (2011) is in fact a crossover but that the original model can be slightly altered in order to give rise to a genuine phase transition, which is of interest on its own. We finally propose a general theoretical framework to understand how a nonlinear impact may appear even in the limit where the bias in the order flow is vanishingly small.