A.1 Derivation of Eq. (3) By expanding Eq. (2) with the definition of εli,t = xli,t µli,t, we have: Et = We note that each xli,t influences Et in two ways: (i) it occurs in Eq. (6) explicitly, but (ii) it also determines the values of µl 1k,t via Eq. Considering also the special cases of l = Land l = 0, we obtain Eq. (3). We note that θl+1i,j affects the value of the function Et of Eq. (6) by influencing µli,t via Eq. Here, we provide further details about training PCNs, useful to reproduce them. Furthermore, we have applied a decay factor of 0.9 to γ, applied each time the energy failed to decrease.

Moreover, the entropyH represents the expressiveness of a deep system, which correlated with the performance of a deep neural network [19]. Note thatCl 1 is equal to 1 when the layer is a depth-wise convolution. According to the work of [19], the input of each layer is zero-mean distribution when deriving the entropy,so that the upper bound ofQisset as2N 1. As we set the quantization step as 1 in Eq. 10, the distribution ofR will be much smoother, and the probability will close to0. Since the Flash budget constrains the total weights of all network layers.

We note that eachxli,t influences Et in two ways: (i) it occurs in Eq.(6) explicitly, but (ii) it also determinesthevaluesof µl 1k,t viaEq.(1). For the experiments on MHNs, the parameterβ was extensively tested, as we have usedβ {1,2,3,5,10,100,1000},and always reported the best result. For experiments comparing against classical Hopfield networks, we have convertedeveryimagetobinary. Results: The results, plotted inFigure 1, are similar tothe ones ofCIFAR10. So far, we analyzed images with Gaussian noise of variance0.2.

Kernel-based hypothesis tests offer a flexible, non-parametric tool to detect high-order interactions in multivariate data, beyond pairwise relationships. Yet the scalability of such tests is limited by the computationally demanding permutation schemes used to generate null approximations. Here we introduce a family of permutation-free high-order tests for joint independence and partial factorisations of $d$ variables. Our tests eliminate the need for permutation-based approximations by leveraging V-statistics and a novel cross-centring technique to yield test statistics with a standard normal limiting distribution under the null. We present implementations of the tests and showcase their efficacy and scalability through synthetic datasets. We also show applications inspired by causal discovery and feature selection, which highlight both the importance of high-order interactions in data and the need for efficient computational methods.

In this paper we present a new algorithm suitable for matching discrete objects such as strings and trees in linear time, thus obviating dynarrtic programming with quadratic time complexity. Furthermore, prediction cost in many cases can be reduced to linear cost in the length of the sequence to be classified, regardless of the number of support vectors. This improvement on the currently available algorithms makes string kernels a viable alternative for the practitioner.

In this paper we present a new algorithm suitable for matching discrete objects such as strings and trees in linear time, thus obviating dynarrtic programming with quadratic time complexity. Furthermore, prediction cost in many cases can be reduced to linear cost in the length of the sequence to be classified, regardless of the number of support vectors. This improvement on the currently available algorithms makes string kernels a viable alternative for the practitioner.

In this paper we present a new algorithm suitable for matching discrete objects such as strings and trees in linear time, thus obviating dynarrtic programming with quadratic time complexity. Furthermore, prediction cost in many cases can be reduced to linear cost in the length of the sequence tobe classified, regardless of the number of support vectors. This improvement on the currently available algorithms makes string kernels a viable alternative for the practitioner.