But it’s obvious that DL and RG are functionally distinct: in the latter, the couplings (i.e., the connection or weight matrix) are fixed by the relationship between the hamiltonian at different scales, while in the former, these connections are dynamically altered in the training process. However, the relation between DL and RG is more subtle than has been previously presented. Note that this is not the cumulant generating function itself, but corresponds to setting therein: Within expectation values, becomes the dimensionless energy , so the moment/cumulant picks up a factor of relative to the usual energetic moments in eqn. From the cumulant expansion in the aforementioned eqn. The real-space RG prescription consists of coarse-graining , and then writing the new distribution in the canonical form (24). Free energy, variational inference, and the brain, Deep learning and the renormalization group. with and as above. operators) as a physical system is examined at different length scales. We compare the ideas behind the RG on the one hand and deep machine learning on the other, where depth and scale play a similar role. %� Deep learning and the renormalization group. ( Log Out /  That’s not to say there aren’t deeper connections between the two: in my earlier post on RBMs for example, I touched on how the cumulants encoding UV interactions appear in the renormalized couplings after marginalizing out hidden degrees of freedom, and we’ll go into this in much more detail below. Accordingly, [8] instead replace (2) with. (1) in the previous post. Fortunately, after banging my head against this for a month, I learned of a recent paper [8] that derives exactly the sort of cumulant relation I was aiming for, at least in the case of generic lattice models. That said, structure is the first step to dynamics, so I wanted to see how far one could push the analogy. Conversely, the neural net directly maps independent Gaussian noises to physical configurations following the inverse RG flow. operators) as a physical system is examined at different length scales. So we instead express the integral — or rather, the discrete sum over lattice sites — in terms of the conditional distribution : where , . Let us now consider sequential marginalizations to obtain and . �. [7]  E. Mello de Koch, R. Mello de Koch, and L. Cheng, “Is Deep Learning an RG Flow?,” arXiv:1906.05212. Copyright © Ro Jefferson. xڅZَ��}�W�[4����C�K�,.� qZTK�k��I�������\4��'��V�:U����>}�U��? This will be of the form (10), where in this case we need to choose , so that, (where, since , the inverse matrix is also invariant under the transpose; at this stage of exploration, I’m being quite cavalier about questions of existence). Note that there are no intra-layer couplings, and that I’ve stacked the layers so that the visible layer is connected only to the intermediate hidden layer , which in turn is connected only to the final hidden layer . A treatment of Principal Component … }���o��� Fill in your details below or click an icon to log in: You are commenting using your WordPress.com account. In analogy with Wilsonian RG, this corresponds to further lowering the cutoff scale in order to obtain a description of the theory in terms of low-energy degrees of freedom that we can actually observe. Change ), You are commenting using your Facebook account. There’s an obvious Bayesian parallel here: we low-energy beings don’t have access to complete information about the UV, so the visible units are naturally identified with IR degrees of freedom, and indeed I’ll use these terms interchangeably throughout. As mentioned above, I’d originally hoped to derive something like a beta function for the cumulants, to see what insights theoretical physics and machine learning might yield to one another at this information-theoretic interface. The first of these is, In order to establish a relationship between couplings at each energy scale, we then define the hamiltonian on the remaining, lower-energy degrees of freedom such that. Finally, by taking the log, we obtain. For example in [2], wouldn’t it have been better to use convoluted RBMs (assuming such a thing exists)? %PDF-1.5 “Dynamics” in this sense are determined by the hamiltonian , which is again one-dimensional. Will it make a considerable difference? ( Log Out /  To start off, I wanted a simple model that would be analytically solvable while making the analogy with decimation RG completely transparent.

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