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Register a new userRG-Flow: A hierarchical and explainable flow model based on renormalization group and sparse prior
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Flow-based generative models have become an important class of unsupervised learning approaches. In this work, we incorporate the key idea of renormalization group (RG) and sparse prior distribution to design a hierarchical flow-based generative model, called RG-Flow, which can separate information at different scales of images with disentangled representations at each scale. We demonstrate our method mainly on the CelebA dataset and show that the disentangled representations at different scales enable semantic manipulation and style mixing of the images. To visualize the latent representations, we introduce receptive fields for flow-based models and find that the receptive fields learned by RG-Flow are similar to those in convolutional neural networks. In addition, we replace the widely adopted Gaussian prior distribution by a sparse prior distribution to further enhance the disentanglement of representations. From a theoretical perspective, the proposed method has $O(log L)$ complexity for image inpainting compared to previous generative models with $O(L^2)$ complexity.
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We apply Feshbach-Krein-Schur renormalization techniques in the hierarchical Anderson model to establish a criterion on the single-site distribution which ensures exponential dynamical localization as well as positive inverse participation ratios and Poisson statistics of eigenvalues. Our criterion applies to all cases of exponentially decaying hierarchical hopping strengths and holds even for spectral dimension $d > 2$, which corresponds to the regime of transience of the underlying hierarchical random walk. This challenges recent numerical findings that the spectral dimension is significant as far as the Anderson transition is concerned.
The gradient flow bears a close resemblance to the coarse graining, the guiding principle of the renormalization group (RG). In the case of scalar field theory, a precise connection has been made between the gradient flow and the RG flow of the Wilson action in the exact renormalization group (ERG) formalism. By imitating the structure of this connection, we propose an ERG differential equation that preserves manifest gauge invariance in Yang--Mills theory. Our construction in continuum theory can be extended to lattice gauge theory.
Casimir energy is calculated for 5D scalar theory in the {it warped} geometry. A new regularization, called {it sphere lattice regularization}, is taken. The regularized configuration is {it closed-string like}. We numerically evaluate $La$(4D UV-cutoff), $om$(5D bulk curvature, extra space UV-boundary parameter) and $T$(extra space IR-boundary parameter) dependence of Casimir energy. 5D Casimir energy is {it finitely} obtained after the {it proper renormalization procedure.} The {it warp parameter} $om$ suffers from the {it renormalization effect}. Regarding Casimir energy as the main contribution to the cosmological term, we examine the dark energy problem.
We determine the global renormalization group (RG) flow of the Sachdev-Ye-Kitaev (SYK) model. This flow allows for an understanding of the surprising role of critical slowing down at a quantum first-order transition in strongly-correlated electronic systems. From a simple truncation of the infinite hierarchy of the exact functional RG flow equations we identify several fixed points: Apart from a stable fixed point, associated with the celebrated non-Fermi liquid state of the model, we find another stable fixed point related to an integer-valence state. These stable fixed points are separated by a discontinuity fixed point with one relevant direction, describing a quantum first-order transition. Most notably, the fermionic spectrum continues to be quantum critical even at the discontinuity fixed point. This rules out a description of this quantum first-order transition in terms of a local effective Ising variable that is established for classical transitions. It reveals that quantum phase coexistence can be a genuine critical state of matter.
We revisit perturbative RG analysis in the replicated Landau-Ginzburg description of the Random Field Ising Model near the upper critical dimension 6. Working in a field basis with manifest vicinity to a weakly-coupled Parisi-Sourlas supersymmetric fixed point (Cardy, 1985), we look for interactions which may destabilize the SUSY RG flow and lead to the loss of dimensional reduction. This problem is reduced to studying the anomalous dimensions of leaders -- lowest dimension parts of $S_n$-invariant perturbations in the Cardy basis. Leader operators are classified as non-susy-writable, susy-writable or susy-null depending on their symmetry. Susy-writable leaders are additionally classified as belonging to superprimary multiplets transforming in particular $textrm{OSp}(d | 2)$ representations. We enumerate all leaders up to 6d dimension $Delta = 12$, and compute their perturbative anomalous dimensions (up to two loops). We thus identify two perturbations (with susy-null and non-susy-writable leaders) becoming relevant below a critical dimension $d_c approx 4.2$ - $4.7$. This supports the scenario that the SUSY fixed point exists for all $3 < d leq 6$, but becomes unstable for $d < d_c$.
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