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    "text": "Backpropagation for Implicit Spectral Densities Aditya Ramesh Yann LeCun\nDepartment of Computer Science, New York University\n{ar2922,yann}@cs.nyu.edu Abstract\n2018 Most successful machine intelligence systems rely on gradient-based learning,\nwhich is made possible by backpropagation. Some systems are designed to aid us\nin interpreting data when explicit goals cannot be provided. These unsupervisedJun systems are commonly trained by backpropagating through a likelihood function. We introduce a tool that allows us to do this even when the likelihood is not\n1 explicitly set, by instead using the implicit likelihood of the model. Explicitly\ndefining the likelihood often entails making heavy-handed assumptions that impede\nour ability to solve challenging tasks. On the other hand, the implicit likelihood of\nthe model is accessible without the need for such assumptions. Our tool, which we\ncall spectral backpropagation, allows us to optimize it in much greater generality\nthan what has been attempted before. GANs can also be viewed as a technique for[cs.LG] optimizing implicit likelihoods. We study them using spectral backpropagation in\norder to demonstrate robustness for high-dimensional problems, and identify two\nnovel properties of the generator G: (1) there exist aberrant, nonsensical outputs to\nwhich G assigns very high likelihood, and (2) the eigenvectors of the metric induced\nby G over latent space correspond to quasi-disentangled explanatory factors. Density estimation is an important component of unsupervised learning. In a typical scenario, we are\ngiven a finite sample D := {x1, . . . , xn}, and must make decisions based on information about the\nhypothetical process that generated D. Suppose that D ∼P, where P is a distribution over some\nsample space X. We can model this generating process by learning a probabilistic model f : Z →X\nthat transforms a simple distribution PZ over a latent space Z into a distribution Q over X.",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
    "authors": [
      "Aditya Ramesh",
      "Yann LeCun"
    ],
    "published_date": "2018-06-01",
    "primary_category": "cs.LG",
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    "text": "When Q is\na good approximation to P, we can use it to generate samples and perform inference. Both operations\nthe model distribution Q. We typically place two important constraints on f in order to make this\npossible: (1) existence of an explicit inverse f −1 : X →Z, and (2) existence of a simple procedure\nby which we can evaluate Q. Often times, f is instead constructed as a map from X to Z in order to\nmake it convenient to evaluate Q(x) for a given observation x ∈X. This is the operation on which\nwe place the greatest demand for throughput during training. Each choice corresponds to making one\nof the two operations – generating samples or evaluating Q – convenient, and the other inconvenient. Regardless of the choice, both operations are required, and so both constraints are made to hold in\npractice.",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
    "authors": [
      "Aditya Ramesh",
      "Yann LeCun"
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    "text": "We regard f as a map from Z to X in this presentation for sake of simplicity. Much of the work in deep probabilistic modeling is concerned with allowing f to be flexible enough\nto capture intricate latent structure, while simultaneously ensuring that both conditions hold. We can\ndichotomize current approaches based on how the second constraint – existence of a simple procedure\nto evaluate Q – is satisfied. The first approach involves making f autoregressive by appropriately\nmasking the weights of each layer. This induces a lower-triangular structure in the Jacobian, since\neach component of the model's output is made to depend only on the previous ones.",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
    "authors": [
      "Aditya Ramesh",
      "Yann LeCun"
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    "published_date": "2018-06-01",
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    "text": "32nd Conference on Neural Information Processing Systems (NIPS 2018), Montréal, Canada. rapidly evaluate the log-determinant term involved in likelihood computation, by accumulating the\ndiagonal elements of the Jacobian. Research into autoregressive modeling dates back several decades, and we only note some recent\ndevelopments. Germain et al. [2015] describe an autoregressive autoencoder for density estimation. Kingma et al. [2016] synthesize autogressive modeling with normalizing flows [Rezende and Mohamed, 2015] for variational inference, and Papamakarios et al. [2017] make further improvements.\nvan den Oord et al. [2016c] apply this idea to image generation, with follow-up work (Dinh et al.\n[2016], van den Oord et al. [2016b]) that exploits parallelism using masked convolutions. van den\nOord et al. [2016a] do the same for audio generation, and van den Oord et al. [2017] introduce\nstrategies to improve efficiency. We refer the reader to Jang [2018] for an excellent overview of these\nworks in more detail.",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
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    "text": "The second approach involves choosing the layers of f to be transformations, not necessarily autoregressive, for which explicit expressions for the Jacobian are still available. We can then evaluate the\nlog-determinant term for f by accumulating the layerwise contributions in accordance with the chain\nrule, using a procedure analogous to backpropagation. Rezende and Mohamed [2015] introduced\nthis idea to variational inference, and recent work, including [Berg et al., 2018] and [Tomczak and\nWelling, 2016], describe new types of such transformations. Both approaches must invariably compromise on model flexibility. An efficient method for differentiating implicit densities that do not fulfill these constraints would enrich the current toolset for\nprobabilistic modeling. Wu et al. [2016] advocate using annealed importance sampling [Neal, 2001]\nfor evaluating implicit densities, but it is not clear how this approach could be used to obtain gradients. Very recent work [Li and Turner, 2017] uses Stein's identity to cast gradient computation for implicit\ndensities as a sparse recovery problem. Our approach, which we call spectral backpropagation,\nharnesses the capabilities of modern automatic differentiation (Abadi et al. [2016], Paszke et al.\n[2017]) by directly backpropagating through an approximation for the spectral density of f. We make the first steps toward demonstrating the viability of this approach by minimizing dKL(Q, PX)\nand dKL(PX, Q), where Q is the implicit density of a non-invertible Wide ResNet [Zagoruyko and\nKomodakis, 2016] f, on a set of test problems. Having done so, we then turn our attention to\ncharacterizing the behavior of the generator G in GANs [Goodfellow et al., 2014], using a series of\ncomputational studies made possible by spectral backpropagation.",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
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    "text": "Our purpose in conducting these\nstudies is twofold. Firstly, we show that our approach is suitable for application to high-dimensional\nproblems. Secondly, we identify two novel properties of generators: • The existence of adversarial perturbations for classification models [Szegedy et al., 2013] is\nparalleled by the existence of aberrant, nonsensical outputs to which G assigns very high\nlikelihood. • The eigenvectors of the metric induced by the G over latent space correspond to meaningful,\nquasi-disentangled explanatory factors. Perturbing latent variables along these eigenvectors\nallows us to quantify the extent to which G makes use of latent space. We hope that these observations will contribute to an improved understanding of how well generators\nare able to capture the latent structure of the underlying data-generating process. 2.1 Generalizing the Change of Variable Theorem We begin by revisiting the geometric intuition behind the usual change of variable theorem. First,\nwe consider a rectangle in R2 with vertices x0, x1, x3, x2 given in clockwise order, starting from the\nbottom-left vertex. To determine its area, we compute its side lengths v1 := x1 −x0, v2 := x2 −x0\nand write V2 = v1v2. Now suppose we are given a parallelepiped in R3 whose sides are described\nby the vectors v1 := x1 −x0, v2 := x2 −x0, and v3 := x3 −x0. Its volume is given by the triple\nproduct V3 = ⟨v1 × v2, v3⟩, where × and ⟨·, ·⟩denote cross product and inner product, respectively. This triple product can be rewritten as V3 = det (v1 v2 v3) ,",
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    "text": "which we can generalize to compute the volume of a parallelepiped in RN: VN = det (v1 · · · vN) . If we regard the vertices x0, . . . , xN as observations in X, the change of variable theorem can be\nunderstood as the differential analog of this formula. To wit, we suppose that f : RN →RN is\na diffeomorphism, and denote by Jf the Jacobian of its output with respect to its input. Now, VN\nbecomes the infinitesimal volume element determined by Jf. For an observation x ∈X, the change\nof variable theorem says that we can compute\nQ(x) = PZ(f −1(x)) |det Jf(f −1(x)))|−1 =: PZ(z) |det(Jf(z))|−1, where we set z := f −1(x). An n-dimensional parallelepiped in RN requires n vectors to specify its sides. When n < N, its\nvolume is given by the more general formula [Hanson, 1994],  ⟨v1, v1⟩ · · · ⟨v1, vn⟩ \nV n2 = det  ... ... ...   .\n⟨vn, v1⟩ · · · ⟨vn, vn⟩",
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    "text": "The corresponding analog of the change of variable theorem is known in the context of geometric\nmeasure theory as the smooth coarea formula [Krantz and Parks, 2008]. When f is a diffeomorphism\nbetween manifolds, it says that Q(x) = PZ(f −1(x)) det(Jf(f −1(x))tJf(f −1(x)))−1/2 =: PZ(z) det Mf(z)−1/2, (1) where we set z := f −1(x) as before, and define Mf := (Jf)tJf to be the metric induced by f over\nthe latent manifold Z. In many cases of interest, such as in GANs, the function f is not necessarily\ninjective. Application of the coarea formula would then require us to evaluate an inner integral\nover {z ∈Z : f(z) = x}, rather than over the singleton {f −1(x)}. We ignore this technicality and\napply Equation 1 anyway.",
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    "text": "The change of variable theorem gives us access to the implicit density Q in the form of the spectral\ndensity of Mf. Indeed, the Lie identity ln det = tr ln allows us to express the log-likelihood\ncorresponding to Equation 1 as\nln Q(x) = ln P(z) −1 ln det Mf(z) = ln P(z) −1 tr ln Mf(z). (2)\n2 2\nWe focus on the factor involving Mf on the RHS, which can be written as tr ln Mf(z) = X ln λ = Eλ∼Pλ ln λ,\nλ∈Sp(Mf (z)) where Sp denotes the spectrum, and Pλ the delta distribution over the eigenvalues in the spectrum. We let θ denote the parameters of f, and assume that PZ is independent of θ. Now, differentiating\nEquation 2 with respect to θ gives Dθ ln Q(x) = −Dθ Eλ∼Pλ ln λ. (3) Equation 3 allows us to formulate gradient computation for implicit densities as a variant of stochastic\nbackpropagation (Kingma and Welling [2013], Rezende et al. [2014]), in which the base distribution\nfor the expectation is the spectral density of Mf rather than a normal distribution. 2.2 An Estimator for Spectral Backpropagation To obtain an estimator for Equation 3, we turn to the thriving literature on stochastic approximation of\nspectral sums. These methods estimate quantities of the form ΣS(A) := tr S(A), where A is a large\nor implicitly-defined matrix, by accessing A using only matrix-vector products. In our case, S = ln,\nand the products involving A = Mf can be evaluated rapidly using automatic differentiation. We\nmake no attempt to conduct a comprehensive survey, but note that among the most promising recent\napproaches are those described by Han et al. [2017], Boutsidis et al. [2017], Ubaru et al. [2017],\nand Fitzsimons et al. [2017].",
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    "text": "Algorithm 1 Procedure to estimate ln det using the Chebyshev approximation. Require: A ∈Rn×n is the implicit matrix; m the desired order; p the number of probe vectors for\nthe trace estimator; t the number of power iterations; g ≥1 a multiplier for the estimate returned\nby the power method; and ϵ the stipulated lower bound on Sp(A).\nprocedure S T O C H A S T I CL O GD E T(A, m, p, t, g, ϵ)\nˆλmax ←P O W E RM E T H O D(A, t)\nµ, ν ←ϵ, g ˆλmax\na, b ←µ/(µ + ν), ν/(µ + ν)\nDefine ϕ and ϕ−1 using Equation 6.\n{ci}i∈[0,m] ←C H E B Y S H E VC O E F F I C I E N T S(ln ◦ϕ)\n¯A ←A/(µ + ν)\nΓ ←S T O C H A S T I CC H E B Y S H E VT R AC E(ϕ−1(¯A), {ci}, p)\nreturn n ln(a + b) + Γ\nprocedure S T O C H A S T I CC H E B Y S H E VT R AC E(A, {ci}i∈[0,m], p)\nr ←0\nfor j ∈[1, p] do\nv ←R A N D O MR A D E M AC H E R(n)\nw0, w1 ←v, Av\ns ←c0w0 + c1w1\nfor i ∈[2, m] do\nwi ←2Awi−1 −wi−2\ns ←s + ciwi\nr ←r + ⟨v, s⟩\nreturn r/p We briefly describe the approaches of Han et al. [2017] and Boutsidis et al. [2017], which work on\nthe basis of polynomial interpolation. Given a function ¯S : [−1, 1] →R, these methods construct an\norder-m approximating polynomial ¯pm to ¯S, given by where ci ∈R and Ti : [−1, 1] →R. The main difference between the two approaches is the choice\nof approximating polynomial. Boutsidis et al. [2017] use Taylor polynomials, for which ¯S(i)\nci := and Ti : x 7→xi, where we use superscript (i) to denote iterated differentiation. On the other hand, Han et al. [2017]\nuse Chebyshev polynomials. These are defined by the recurrence relation T0 = 1, T1 : x 7→x, and Ti : x 7→2xTi−1(x) −Ti−2(x), i ≥2. (4) The coefficients {ci} for the Chebyshev polynomials are called the Chebyshev nodes, and are defined  1 X ¯S(xj)T0(xj), i = 0,\nm + 1\nj∈[0,m]  ci :=\nX ¯S(xj)Ti(xj), i ≥1.\nm + 1\nj∈[0,m]  Now suppose that we are given a matrix ¯A ∈Rn×n such that Sp(¯A) ⊂[−1, 1].",
    "paper_id": "1806.00499",
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    "text": "After having made a\nchoice for the construction of ¯pm, we can use the approximation Σ ¯S(¯A) = tr ¯S(¯A) = X ¯S(λ) ≈ X ¯pm(λ)\nλ∈Sp( A)¯ λ∈Sp( A)¯\n= X X ciTi(λ) = X ci X Ti(λ)\nλ∈Sp( A)¯ i∈[0,m] i∈[0,m] λ∈Sp( A)¯\n= X ci tr Ti(¯A). (5) This reduces the problem of estimating the spectral sum Σ ¯S(¯A) to computing the traces tr Ti(¯A) for\nall i ∈[0, m]. Two issues remain in applying this approximation. The first is that both dom(¯S) and Sp(¯A) are\nrestricted to [−1, 1]. In our case, ln : (0, ∞) →R, and Mf(z) can be an arbitrary positive definite\nmatrix. To address this issue, we define ϕ : [−1, 1] →[a, b], where\nϕ : x 7→b −a x + b + a and ϕ−1 : x 7→ 2 −b + a (6) 2 2 b −ax b −a. Now we set ¯S := S ◦ϕ, so that S = ¯S ◦ϕ−1 ≈¯pm ◦ϕ−1 = X ci(Ti ◦ϕ−1) =: pm. We stress that while pm is defined using ϕ−1, the coefficients ci are computed using ¯S := S ◦ϕ. With these definitions in hand, we can write\nΣS(¯A) = Σ¯S◦ϕ−1(¯A) ≈tr pm(¯A). (7) Han et al. [2017] require spectral bounds µ and ν, so that Sp(A) ⊂[µ, ν], and set µ ν A\na := , b := , and ¯A := µ + ν µ + ν a + b. After using Equation 7 with a Chebyshev approximation for S = ln to obtain Γ ≈ln det(¯A), we\ncompute\nln det(A) = n ln(a + b) + ln det(¯A) ≈n ln(a + b) + Γ. Boutsidis et al. [2017] instead define B := A/ν, and write",
    "paper_id": "1806.00499",
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    "authors": [
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    "text": "ln det(B) = tr ln(B) = tr ln(I −(I −B)). This time, we set ¯A = I −B and use Equation 7 with a Taylor approximation for S = ln(1 −x) to\nobtain Γ ≈ln det(B). Then, we compute\nln det(A) = n ln ν + ln det(¯A) ≈n ln ν + Γ. We can easily obtain an accurate upper bound ν using the power method. The lower bound µ is fixed\nto a small, predetermined constant in our work. The second issue is that deterministically evaluating the terms tr Ti(¯A) in Equation 5 requires us\nto compute matrix powers of ¯A. Thankfully, we can drastically reduce the computational cost and\napproximate these terms using only matrix-vector products.",
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    "text": "This is made possible by the stochastic\ntrace estimator introduced by Hutchinson [1990]: X ⟨vj, ¯Avj⟩. tr ¯A = Ev∼PV ⟨v, ¯Av⟩≈1\nj∈[1,p] When the distribution PV for the probe vectors v has expectation zero, the estimate is unbiased. We\nuse the Rademacher distribution, which samples the components of v uniformly from {−1, 1}. We\nrefer the reader to Avron and Toledo [2011] for a detailed study on the variance of this estimator. (b) Samples from f for\nthe four test energies at\nepochs 5, 10, 9, and 4, respectively. (c) Samples from f for the\nlast two test energies, both at\nepoch 26.\n(a) Plots of the loss and relative error for minimizing dKL(Q, P). Figure 1: Results for minimizing dKL(Q, PX) for the four test energies described in Rezende and\nMohamed [2015]. Subfigures (b) and (c) show the model samples superimposed over contour plots\nof the corresponding ground-truth test energies. Each epoch corresponds to 5000 iterations.",
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    "text": "We see\nin (a) that the relative error for the approximation to the log determinant typically stays below 30%,\nexcept toward the end of training for the last two test energies. At this point, samples from these two\nmodels begin to drift away from the origin, as shown in (c). We first describe how the trace estimator is applied when a Taylor approximation is used to construct ¯pm. In this case, we have",
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    "text": "* + 1 tr ¯pm(¯A) = X ci tr ¯Ai ≈1 X ci X ⟨vj, ¯Aivj⟩= X ci vj, X ¯Aivj .\np p\ni∈[0,m] i∈[0,m] j∈[1,p] j∈[1,p] i∈[0,m]\nThe inner summands ¯Aivj are evaluated using the recursion w0 := vj and wi := ¯Awi−1 for i ≥1. It\nfollows that the number of matrix-vector products involved in the approximation increases linearly\nwith respect to the order m of the approximating polynomial ¯pm. The same idea allows us to\naccumulate the traces for the Chebyshev approximation, based on Equation 4. The resulting procedure\nis given in Algorithm 1; it is our computational workhorse for evaluating the log-likelihood in\nEquation 2 and estimating the gradient in Equation 3. 3 Learning Implicit Densities Suppose that we are tasked with matching a given data distribution PX with the implicit density Q of\nthe model f. Two approaches for learning f are available, and the choice of which to use depends on\nthe type of access we have to the data distribution PX. The first approach – minimizing dKL(Q, PX) –\nis applicable when we know how to evaluate the likelihood of PX, but are not necessarily able to\nsample from it. The second approach – minimizing dKL(PX, Q) – is applicable when we are able\nto sample from PX, but are not necessarily able to evaluate its likelihood. We show that spectral\nbackpropagation can be used in both cases, when neither of the two conditions described in Section 1\nholds. All of the examples considered here are densities over R2. We match them by transforming a prior PZ\ngiven by a spherical normal distribution.",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
    "authors": [
      "Aditya Ramesh",
      "Yann LeCun"
    ],
    "published_date": "2018-06-01",
    "primary_category": "cs.LG",
    "arxiv_url": "http://arxiv.org/abs/1806.00499v1",
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    "text": "Our choice for the architecture of f is a Wide ResNet\ncomprised of four residual blocks. Each block is a three-layer bottleneck from R2 to R2 whose hidden\nlayer size is 32. All layers are equipped with biases, and use LeakyReLU activations.",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
    "authors": [
      "Aditya Ramesh",
      "Yann LeCun"
    ],
    "published_date": "2018-06-01",
    "primary_category": "cs.LG",
    "arxiv_url": "http://arxiv.org/abs/1806.00499v1",
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    "text": "(b) Contour plots of the loglikelihood of f for the two\ntest distributions, at the end of\ntraining. (a) Plots for the loss and relative error for minimizing dKL(P, Q). Figure 2: Results for minimizing dKL(PX, Q) for the crescent and circular mixture densities whose\ndefinitions are given in the supplementary material. The relative error for the approximation to the log\ndeterminant typically stays below 5%. In (b), we show samples from PX superimposed over contour\nplots of the log-likelihood of f.",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
    "authors": [
      "Aditya Ramesh",
      "Yann LeCun"
    ],
    "published_date": "2018-06-01",
    "primary_category": "cs.LG",
    "arxiv_url": "http://arxiv.org/abs/1806.00499v1",
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    "text": "the gradient updates using a batch size of 64, and apply the updates using Adam [Kingma and Ba,\n2014] with a step size of 1 × 10−4, and all other parameters kept at their the default values. To compute the gradient update given by Equation 3, we use Algorithm 1 with (m, p, t, g) :=\n(10, 20, 20, 1.2) for all experiments. For minimizing dKL(Q, PX), we use ϵ := 0.1, and for minimizing dKL(PX, Q), ϵ := 1 × 10−2. In order to monitor the accuracy of the approximation for the\nlikelihood, we compute Mf at each iteration, and evaluate the ground-truth likelihood in accordance\nwith Equation 2. We define the relative error of the approximation ln ˆℓwith respect to the ground-truth\nlog-likelihood ln ℓby ˆℓ ˆℓ−ℓ ! ˆℓ−ℓ\nln ˆℓ−ln ℓ= ln ℓ= ln 1 + ℓ ≈ ℓ , provided that the quotient is not too large. This definition of relative error avoids numerical problems\nwhen ℓ≈0.",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
    "authors": [
      "Aditya Ramesh",
      "Yann LeCun"
    ],
    "published_date": "2018-06-01",
    "primary_category": "cs.LG",
    "arxiv_url": "http://arxiv.org/abs/1806.00499v1",
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    "text": "We begin by considering the first approach, in which we seek to minimize dKL(Q, PX). This objective\nrequires that we be able to sample from Q, so we choose f : Z →X to be a map from latent space to\nobservation space. The results are shown in Figure 1. To prevent f from making Q collapse to model\nthe infinite support of PX, we found it helpful to incorporate the regularizer into the objectives for the third and fourth test energies. Here, ∥·∥2 denotes the spectral norm. To\nimplement this regularizer, we simply backpropagate through the estimate of λmax that is already\nproduced by Algorithm 1. We use ρ := 8 × 10−2 in both cases. Despite the use of this regularizer, we\nfind that continuing to train the models for these last two test energies causes the samples to drift away\nfrom the origin (see Figure 1(c)). We have not made any attempt to address this behavior. Finally, we\nnote that since dKL(Q, PX) is bounded from below by the negative log-normalization constant of Q,\nit can become negative when Q is unnormalized. We see that this happens for all four examples.",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
    "authors": [
      "Aditya Ramesh",
      "Yann LeCun"
    ],
    "published_date": "2018-06-01",
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    "text": "In the second approach, we seek to minimize dKL(PX, Q). This objective requires that we be able to\nevaluate the likelihood of Q, so we choose f : X →Z to be a map from observation space to latent\nspace. The results are shown in Figure 2. We note that minimizing dKL(PX, Q) is ill-posed when Q\nis unnormalized. In this scenario, the model distribution Q can match PX while also assigning mass\noutside the support of PX. We see that this expected behavior manifests in both examples.",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
    "authors": [
      "Aditya Ramesh",
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    "published_date": "2018-06-01",
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    "text": "4 Evaluating GAN Likelihoods For our explorations involving GANs, we train a series of DCGAN [Radford et al., 2015] models\non 64 × 64 rescaled versions of the CelebA [Liu et al., 2015] and LSUN Bedroom datasets. We vary\nmodel capacity in terms of the base feature map count multiplier nf for the DCGAN architecture. (b) Contour plots of the loglikelihoods evaluated over the\nimage grids to the left.",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
    "authors": [
      "Aditya Ramesh",
      "Yann LeCun"
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    "published_date": "2018-06-01",
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    "text": "(a) Samples from the CelebA model with nz = 128, nf = 64 (left) and the\nLSUN Bedroom model with nz = 256, nf = 64 (right). Figure 3: Samples from two models evaluated at fixed grids in latent space (left), and contour plots of\nthe model log-likelihood evaluated over the same grids (right). (a) Trajectory at iterations 0, 100, 500, 1000, 2000, (b) Trajectory at iterations 0, 50, 100, 250, 500, 1000\nand 3000 (3500 total). (1000 total). (c) Trajectory at iterations 0, 100, 500, 1000, 2500, (d) Trajectory at iterations 0, 250, 500, 1000, 2500,\n5000 (5000 total). 5000 (5000 total). Trial log(pfinal/pinit) Initial (λmin, λmax) Final (λmin, λmax) (m, p, t, g, ϵ) (a) 161.40 (2.31 × 10−1, 2.91 × 103) (8.11 × 10−3, 1.06 × 103) (5, 20, 20, 1.1, 1 × 10−4)\n(b) 667.43 (2.71 × 10−1, 2.23 × 103) (2.51 × 10−6, 2.57 × 10−1) (5, 20, 20, 1.1, 1 × 10−5)\n(c) 90.83 (1.87 × 10−2, 2.00 × 103) (8.58 × 10−3, 5.62 × 102) (5, 10, 10, 1.2, 1 × 10−2)\n(d) 336.07 (2.29 × 10−2, 1.24 × 103) (6.12 × 10−4, 4.04 × 102) (5, 10, 10, 1.2, 1 × 10−2)",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
    "authors": [
      "Aditya Ramesh",
      "Yann LeCun"
    ],
    "published_date": "2018-06-01",
    "primary_category": "cs.LG",
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    "text": "Figure 4: Trajectories of four latent variables as they are perturbed to optimize likelihood under\nthe generator. Trajectories (a) and (b) correspond to the CelebA model with nz = 128, nf = 64,\nand trajectories (c) and (d) to the LSUN Bedroom model with nz = 256, nf = 64. Statistics from\nthese trajectories are tabulated above. The last column of the table specifies the parameters used for\nAlgorithm 1 to compute the gradient estimates.",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
    "authors": [
      "Aditya Ramesh",
      "Yann LeCun"
    ],
    "published_date": "2018-06-01",
    "primary_category": "cs.LG",
    "arxiv_url": "http://arxiv.org/abs/1806.00499v1",
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    "text": "generator and discriminator have five layers each, and use translated LeakyReLU activations [Xiang\nand Li, 2017]. To stabilize training, we use weight normalization with fixed scale factors in the\ndiscriminator [Salimans et al., 2016]. Our prior is defined by PZ := unif([−1, 1])nz, where nz is the\nsize of the embedding space. All models were trained for 750 000 iterations with a batch size of 32,\nusing RMSProp with step size 1 × 10−4 and decay factor 0.9. We present results from two of these\nmodels in Figure 3. We apply spectral backpropagation to explore the effect of perturbing a given latent variable z ∈Z\nto maximize likelihood under the generator distribution Q. This is readily accomplished by noting\nthat the same procedure to evaluate Equation 3 can also be used to obtain gradients with respect to z. The results are shown in Figure 4. Intuitively, we might expect the outputs to be transformed in such\na way that they gravitate towards modes of the dataset.",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
    "authors": [
      "Aditya Ramesh",
      "Yann LeCun"
    ],
    "published_date": "2018-06-01",
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    "text": "But this is not what happens. Instead, the\noutputs are transformed into highly aberrant, out-of-distribution examples while nonetheless attaining\nvery high likelihood. As optimization proceeds, Mf also becomes increasingly ill-conditioned. This\nshows that likelihood for generators need not correspond to intuitive notions of visual plausibility. (a) Log spectra for CelebA models with nf = 64 and nz varied. (b) Log spectra for LSUN Bedroom models with nf = 256 and nz varied. (c) CelebA (nz = 32, nf = 64), trials 10 and 11. (d) LSUN Bedroom (nz = 128, nf = 64), trials\nStep size: 0.40. 5 and 12. (e) CelebA (nz = 64, nf = 64), trials 6 and 12. (f) LSUN Bedroom (nz = 256, nf = 64), trials 6\nStep size: 0.65. and 8. (g) Trials 7 and 9 (nz = 128, nf = 64). (h) Trials 8 and 10 (nz = 512, nf = 64).",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
    "authors": [
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    "published_date": "2018-06-01",
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    "text": "Figure 5: Result of perturbing latent variables along the eigenvectors of Mf. In (a) and (b), we show\nthe top eigenvalues of Sp(Mf) evaluated at 12 trial latent variables. Small eigenvalues are not shown.",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
    "authors": [
      "Aditya Ramesh",
      "Yann LeCun"
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    "published_date": "2018-06-01",
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    "text": "The leftmost images of each grid both contain duplicates of the original, each corresponding one\nof the 12 latent variables. The top row shows the effect of applying perturbations along random\ndirections, and the bottom row the result of applying perturbations with the same step size along the\neigenvectors. 5 Uncovering Latent Explanatory Factors The generator in GANs is well-known for organizing latent space such that semantic features can be\ntransferred by means of algebraic operations over latent variables [Radford et al., 2015]. This suggests\nthe existence of a systematic organization of latent space, but perhaps one that cannot be globally\ncharacterized in terms of a handful of simple explanatory factors. We instead explore whether local\nchanges in latent space can be characterized in this way. Since the metric Mf describes local change\nin the generator's output, it is natural to consider the effect of perturbations along its eigenvectors. To\nthis end, we fix 12 trial embeddings in latent space, and compare the effect of perturbations along",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
    "authors": [
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    "text": "(a) CelebA (b) LSUN Bedroom (c) CelebA (d) LSUN Bedroom\n(nf = 64, nz varied). (nf = 64, nz varied). (nz = 128, nf varied). (nz = 256, nf varied). Dataset veff(τ = 1) for nf fixed, nz varied veff(τ = 1) for nz fixed, nf varied\n(nz/2, nf ) (nz, nf ) (2nz, nf ) (nz, nf /2) (nz, nf ) (nz, 2nf ) CelebA (nz = 64, nf = 64) 6.95 10.49 16.55 16.14 16.55 17.29\nLSUN Bedroom (nz = 64, nf = 128) 25.23 34.93 53.60 28.37 34.93 40.04 (e) Effect of varying nz and nf on veff(τ = 1). Figure 6: Effect of varying nz and nf on veff(τ). Plots (a)–(d) show veff as a function of τ. Larger\nvalues for veff suggest increased utilization of latent space. Table (e) shows that doubling nz roughly\ndoubles veff. On the other hand, doubling nf does not result in noticeable change for CelebA, and\nonly results in a modest increase in veff for LSUN Bedroom.",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
    "authors": [
      "Aditya Ramesh",
      "Yann LeCun"
    ],
    "published_date": "2018-06-01",
    "primary_category": "cs.LG",
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    "text": "The step sizes used for the perturbations\nare the same as those reported in Figure 5. random directions to perturbations along these eigenvectors. The random directions are obtained by\nsampling from a spherical normal distribution. We show the results in Figure 5. We can see that dominant eigenvalues, especially the principal eigenvalue, often result in the most\ndrastic changes. Furthermore, these changes are not only semantically meaningful, but also tend\nto make modifications to distinct attributes of the image. To see this more clearly, we consider the\ntop two rows of Figure 5(g). Movement along the first two eigenvectors changes hair length and\nfacial orientation; movement along the third eigenvector decreases the length of the bangs; movement\nalong the fourth and fifth eigenvectors changes background color; and movement along the sixth and\nseventh eigenvectors changes hair color.",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
    "authors": [
      "Aditya Ramesh",
      "Yann LeCun"
    ],
    "published_date": "2018-06-01",
    "primary_category": "cs.LG",
    "arxiv_url": "http://arxiv.org/abs/1806.00499v1",
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    "text": "Inspecting the two columns (c), (e), (g), and (d), (f), (h) in Figure 5 suggests that larger values of nz\nmay encourage the generator to capture more explanatory factors, possibly at the price of decreased\nsample quality. We would like to explore the effect of varying nz and nf on the number of such\nfactors. To do this, we fix a sample of latent variables S := {zj}j∈[1,M] ∼PZ. For each zj ∈S, we\ndefine\n∥G(zj + αv(j)i ) −G(zj)∥2\nδ(j, 0) := Eϵ∼N(0,I)∥G(zj + αϵ) −G(zj)∥2 and δ(j, i) := ,\nδ(j, 0)\nfor every eigenvector v(j)1 , . . . , v(j)nz of Mf(zj). The quantity δ(j, i) measures the pixelwise change\nresulting from a perturbation along an eigenvector, relative to the change we expect from a random\nperturbation. Finally, we define\nveff(τ) := X X 1{δ(j, i) > τ},\nj∈[1,M] i∈[1,nz] where 1{·} is the indicator function. This quantity measures the average number of eigenvectors for\nwhich the relative change is greater than the threshold τ. As such, it can be regarded as an effective\nmeasure of dimensionality for latent space. We explore the effect of varying nz and nf on veff in\nFigure 6. Current approaches for probabilistic modeling attempt to satisfy two goals that are fundamentally at\nodds with one another: fulfillment of the two constraints described in Section 1, and model flexibility. In this work, we develop a computational tool that aims to expand the scope of probabilistic modeling\nto functions that do not satisfy these constraints. We make the first steps toward demonstrating\nfeasibility of this approach by minimizing divergences in far greater generality than what has been\nattempted before. Finally, we uncover surprising facts about the organization of latent space for GANs\nthat we hope will contribute to an improved understanding of how effectively they capture underlying\nlatent structure.",
    "paper_id": "1806.00499",
    "title": "Backpropagation for Implicit Spectral Densities",
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