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"content": "In this work, we explore the theoretical properties of conditional deep generative models under the statistical framework of distribution regression where the response variable lies in a high-dimensional ambient space but concentrates around a potentially lower-dimensional manifold. More specifically, we study the large-sample properties of a likelihood-based approach for estimating these models. Our results lead to the convergence rate of a sieve maximum likelihood estimator (MLE) for estimating the conditional distribution (and its devolved counterpart) of the response given predictors in the Hellinger (Wasserstein) metric. Our rates depend solely on the intrinsic dimension and smoothness of the true conditional distribution. These findings provide an explanation of why conditional deep generative models can circumvent the curse of dimensionality from the perspective of statistical foundations and demonstrate that they can learn a broader class of nearly singular conditional distributions. Our analysis also emphasizes the importance of introducing a small noise perturbation to the data when they are supported sufficiently close to a manifold. Finally, in our numerical studies, we demonstrate the effective implementation of the proposed approach using both synthetic and real-world datasets, which also provide complementary validation to our theoretical findings."
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"content": "The problem of non-parametric conditional density estimation has been extensively explored in statistical literature. Hall & Yao (2005), Bott & Kohler (2017), and Bilodeau et al. (2023) directly tackle this problem with smoothing and local polynomial-based methods. Fan & Yim (2004) and Efromovich (2007) explore suitably transformed regression problems to address this challenge. Other notable approaches include the nearest neighbor method (Izbicki et al., 2020; Bhattacharya & Gangopadhyay, 1990), basis function expansion (Sugiyama et al., 2010; Izbicki & Lee, 2016), tree-based boosting (Pospisil & Lee, 2018; Gao & Hastie, 2022), and Bayesian optimal transport flow Chemseddine et al. (2024) among others."
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"content": "In the context of conditional generation, we highlight recent work by Zhou et al. (2022) and Liu et al. (2021). In Zhou et al. (2022), GANs were employed to investigate conditional density estimation. While this work offers a consistent estimator, it lacks statistical rates or convergence analysis, and its focus is on a low-dimensional setup. In Liu et al. (2021), conditional density estimation supported on a manifold using Wasserstein-GANs was examined. However, their setup does not account for smoothness across either covariates or responses, nor do they address how deep generative models specifically tackle the challenges of high-"
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"content": "dimensionality. Moreover, their assumption that the data lies exactly on the manifold can be restrictive. Our study shares some commonalities with the work of Chae et al. (2023), as both investigate sieve maximum likelihood estimators (MLEs). However, the fundamental problems addressed and the methodologies employed differ significantly, and our work involves technical challenges that span multiple scales. While Chae et al. (2023) concentrates exclusively on unconditional distribution estimation, our theoretical analysis necessitates much more nuanced techniques due to the conditional nature of our setup. This shift is noteworthy because it demands a more refined analysis of entropy bounds, considering two potential sources of smoothness - across the regressor and the response variables. Furthermore, our setting accommodates the possibility of an infinite number of "
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"content": ". While "
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"type": "inline_equation",
"content": "f \\in \\mathcal{H}^\\beta((-K, K)^d, K)"
},
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"type": "text",
"content": ", its intrinsic dimension is "
},
{
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],
"type": "inline_equation",
"content": "t_* = 1"
},
{
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],
"type": "text",
"content": " with intrinsic smoothness "
},
{
"bbox": [
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],
"type": "inline_equation",
"content": "\\beta"
},
{
"bbox": [
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],
"type": "text",
"content": ". This mitigates the curse of dimensionality."
}
]
}
],
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],
"type": "text",
"angle": 0,
"lines": [
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"type": "text",
"content": "Example (One-dimensional "
},
{
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518
],
"type": "inline_equation",
"content": "\\beta"
},
{
"bbox": [
52,
480,
290,
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],
"type": "text",
"content": "-Hölder Generator). Let "
},
{
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],
"type": "inline_equation",
"content": "U \\sim \\mathrm{Unif}(0,1)"
},
{
"bbox": [
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],
"type": "text",
"content": " and define "
},
{
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],
"type": "inline_equation",
"content": "G(u) = u^{1/\\beta}"
},
{
"bbox": [
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],
"type": "text",
"content": ", "
},
{
"bbox": [
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],
"type": "inline_equation",
"content": "u \\in [0,1]"
},
{
"bbox": [
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],
"type": "text",
"content": ". Then "
},
{
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],
"type": "inline_equation",
"content": "X = G(U)"
},
{
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],
"type": "text",
"content": " has density"
}
]
}
],
"index": 7
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"type": "interline_equation",
"angle": 0,
"lines": [
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"spans": [
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"bbox": [
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"type": "interline_equation",
"content": "\\frac {d}{d x} \\mathbb {P} (U \\leq x ^ {\\beta}) = \\beta x ^ {\\beta - 1}, \\quad x \\in [ 0, 1 ],",
"image_path": "2fefaa25edad2e8acf5953929f9aa7b46a16fe5b7c9478203f098a0a3499a219.jpg"
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]
}
],
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"type": "text",
"angle": 0,
"lines": [
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"spans": [
{
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],
"type": "text",
"content": "which belongs to the "
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{
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],
"type": "inline_equation",
"content": "\\beta"
},
{
"bbox": [
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291,
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],
"type": "text",
"content": "-Hölder class on "
},
{
"bbox": [
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],
"type": "inline_equation",
"content": "[0,1]"
},
{
"bbox": [
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],
"type": "text",
"content": ". In our notation one checks "
},
{
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],
"type": "inline_equation",
"content": "t_* = 1, \\beta_* = \\beta"
},
{
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],
"type": "text",
"content": ", and setting "
},
{
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],
"type": "inline_equation",
"content": "\\beta = 1"
},
{
"bbox": [
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],
"type": "text",
"content": " recovers Unif(0,1) case and thus provides a fully explicit illustration of Assumption 2."
}
]
}
],
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},
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"type": "text",
"angle": 0,
"lines": [
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"bbox": [
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],
"spans": [
{
"bbox": [
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],
"type": "text",
"content": "Assumption 3. Let "
},
{
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],
"type": "inline_equation",
"content": "\\mathcal{M}_*"
},
{
"bbox": [
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],
"type": "text",
"content": " be the closure of "
},
{
"bbox": [
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],
"type": "inline_equation",
"content": "G_*(\\mathcal{Z},\\mathcal{X})"
},
{
"bbox": [
52,
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291,
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],
"type": "text",
"content": ". We assume that "
},
{
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],
"type": "inline_equation",
"content": "\\mathcal{M}_*"
},
{
"bbox": [
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291,
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],
"type": "text",
"content": " does not have an interior point, and reach "
},
{
"bbox": [
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],
"type": "inline_equation",
"content": "(\\mathcal{M}_*) = \\mathsf{r}_*"
},
{
"bbox": [
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],
"type": "text",
"content": " with "
},
{
"bbox": [
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],
"type": "inline_equation",
"content": "\\mathsf{r}_* > 0"
},
{
"bbox": [
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291,
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],
"type": "text",
"content": "."
}
]
}
],
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},
{
"bbox": [
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],
"type": "text",
"angle": 0,
"lines": [
{
"bbox": [
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"spans": [
{
"bbox": [
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"type": "text",
"content": "Assumption 2 permits low intrinsic dimensionality within the learnable function class. Assumption 3 imposes the strong identifiability condition necessary for efficient estimation, as seen in manifold literature (Aamari & Levrard, 2019; Tang & Yang, 2023)."
}
]
}
],
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},
{
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],
"type": "text",
"angle": 0,
"lines": [
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"bbox": [
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"spans": [
{
"bbox": [
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],
"type": "text",
"content": "Given two conditional densities "
},
{
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],
"type": "inline_equation",
"content": "p_1(\\cdot |x), p_2(\\cdot |x)"
},
{
"bbox": [
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],
"type": "text",
"content": " and "
},
{
"bbox": [
304,
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],
"type": "inline_equation",
"content": "\\mu_X^*"
},
{
"bbox": [
304,
67,
542,
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],
"type": "text",
"content": " denoting the density of "
},
{
"bbox": [
304,
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542,
200
],
"type": "inline_equation",
"content": "X"
},
{
"bbox": [
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],
"type": "text",
"content": ", we use integrated distances for a measure of evaluation. With a slight abuse of notation, we denote "
},
{
"bbox": [
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],
"type": "inline_equation",
"content": "d_1(p_1, p_2) = \\mathbb{E}_X[d_1(p_1(\\cdot |x), p_2(\\cdot |x))]"
},
{
"bbox": [
304,
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],
"type": "text",
"content": " and "
},
{
"bbox": [
304,
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200
],
"type": "inline_equation",
"content": "d_H(p_1, p_2) = \\mathbb{E}_X[d_H(p_1(\\cdot |x), p_2(\\cdot |x)]"
},
{
"bbox": [
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200
],
"type": "text",
"content": ", where "
},
{
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200
],
"type": "inline_equation",
"content": "d_1"
},
{
"bbox": [
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],
"type": "text",
"content": " and "
},
{
"bbox": [
304,
67,
542,
200
],
"type": "inline_equation",
"content": "d_H"
},
{
"bbox": [
304,
67,
542,
200
],
"type": "text",
"content": " represent the "
},
{
"bbox": [
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67,
542,
200
],
"type": "inline_equation",
"content": "L_1"
},
{
"bbox": [
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67,
542,
200
],
"type": "text",
"content": " and the Hellinger distance as "
},
{
"bbox": [
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],
"type": "inline_equation",
"content": "d_1(p_1(\\cdot |x), p_2(\\cdot |x)) = \\int |p_1(y|x) - p_2(y|x)| dy"
},
{
"bbox": [
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],
"type": "text",
"content": " and "
},
{
"bbox": [
304,
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],
"type": "inline_equation",
"content": "d_H(p_1, p_2) = (\\int \\int [\\sqrt{p_1(y|x)} - \\sqrt{p_2(y|x)}]^2 dy)^{1/2}"
},
{
"bbox": [
304,
67,
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200
],
"type": "text",
"content": " respectively. Denote "
},
{
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200
],
"type": "inline_equation",
"content": "\\mathcal{N}(\\delta, \\mathcal{F}, d)"
},
{
"bbox": [
304,
67,
542,
200
],
"type": "text",
"content": " and "
},
{
"bbox": [
304,
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],
"type": "inline_equation",
"content": "\\mathcal{N}_{\\mathbb{I}}(\\delta, \\mathcal{F}, d)"
},
{
"bbox": [
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],
"type": "text",
"content": " as covering and bracketing numbers of the function class "
},
{
"bbox": [
304,
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542,
200
],
"type": "inline_equation",
"content": "\\mathcal{F}"
},
{
"bbox": [
304,
67,
542,
200
],
"type": "text",
"content": " with respect to the (pseudo)-metric "
},
{
"bbox": [
304,
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200
],
"type": "inline_equation",
"content": "d"
},
{
"bbox": [
304,
67,
542,
200
],
"type": "text",
"content": "."
}
]
}
],
"index": 12
},
{
"bbox": [
304,
206,
542,
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],
"type": "text",
"angle": 0,
"lines": [
{
"bbox": [
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],
"spans": [
{
"bbox": [
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],
"type": "text",
"content": "We first present Lemma 1, which establishes the bracketing entropy of the functional class "
},
{
"bbox": [
304,
206,
542,
266
],
"type": "inline_equation",
"content": "\\mathcal{P}"
},
{
"bbox": [
304,
206,
542,
266
],
"type": "text",
"content": " with respect to Hellinger distance in terms of the covering entropy of the search class "
},
{
"bbox": [
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206,
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],
"type": "inline_equation",
"content": "\\mathcal{F}"
},
{
"bbox": [
304,
206,
542,
266
],
"type": "text",
"content": ". This enables us to transfer the entropy control of the individual components "
},
{
"bbox": [
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206,
542,
266
],
"type": "inline_equation",
"content": "\\mathcal{F}"
},
{
"bbox": [
304,
206,
542,
266
],
"type": "text",
"content": " and "
},
{
"bbox": [
304,
206,
542,
266
],
"type": "inline_equation",
"content": "\\sigma"
},
{
"bbox": [
304,
206,
542,
266
],
"type": "text",
"content": " to the entire "
},
{
"bbox": [
304,
206,
542,
266
],
"type": "inline_equation",
"content": "\\mathcal{P}"
},
{
"bbox": [
304,
206,
542,
266
],
"type": "text",
"content": "."
}
]
}
],
"index": 13
},
{
"bbox": [
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269,
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],
"type": "text",
"angle": 0,
"lines": [
{
"bbox": [
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269,
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342
],
"spans": [
{
"bbox": [
304,
269,
542,
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],
"type": "text",
"content": "Lemma 1. Let "
},
{
"bbox": [
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269,
542,
342
],
"type": "inline_equation",
"content": "\\mathcal{F}"
},
{
"bbox": [
304,
269,
542,
342
],
"type": "text",
"content": " be class of functions from "
},
{
"bbox": [
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269,
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],
"type": "inline_equation",
"content": "\\mathcal{Z} \\times \\mathcal{X}"
},
{
"bbox": [
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269,
542,
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],
"type": "text",
"content": " to "
},
{
"bbox": [
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269,
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],
"type": "inline_equation",
"content": "\\mathbb{R}^D"
},
{
"bbox": [
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269,
542,
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],
"type": "text",
"content": " such that "
},
{
"bbox": [
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],
"type": "inline_equation",
"content": "\\| |g|_{\\infty}\\|_{\\infty} \\leq K"
},
{
"bbox": [
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269,
542,
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],
"type": "text",
"content": " for every "
},
{
"bbox": [
304,
269,
542,
342
],
"type": "inline_equation",
"content": "g \\in \\mathcal{F}"
},
{
"bbox": [
304,
269,
542,
342
],
"type": "text",
"content": ". Let "
},
{
"bbox": [
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269,
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],
"type": "inline_equation",
"content": "\\mathcal{P} = \\{P_{g,\\sigma}: g \\in \\mathcal{F}, \\sigma \\in [\\sigma_{\\min}, \\sigma_{\\max}]\\}"
},
{
"bbox": [
304,
269,
542,
342
],
"type": "text",
"content": " with "
},
{
"bbox": [
304,
269,
542,
342
],
"type": "inline_equation",
"content": "\\sigma_{\\min} \\leq 1"
},
{
"bbox": [
304,
269,
542,
342
],
"type": "text",
"content": ". Then, there exist constants "
},
{
"bbox": [
304,
269,
542,
342
],
"type": "inline_equation",
"content": "c = c(\\sigma_{\\max}, K, D)"
},
{
"bbox": [
304,
269,
542,
342
],
"type": "text",
"content": " and "
},
{
"bbox": [
304,
269,
542,
342
],
"type": "inline_equation",
"content": "C = C(\\sigma_{\\max}, K, D)"
},
{
"bbox": [
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269,
542,
342
],
"type": "text",
"content": " and "
},
{
"bbox": [
304,
269,
542,
342
],
"type": "inline_equation",
"content": "\\delta_* = \\delta_*(D)"
},
{
"bbox": [
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269,
542,
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],
"type": "text",
"content": " such that for every "
},
{
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542,
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],
"type": "inline_equation",
"content": "\\delta \\in (0, \\delta_*)"
},
{
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342
],
"type": "text",
"content": ","
}
]
}
],
"index": 14
},
{
"bbox": [
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],
"type": "interline_equation",
"angle": 0,
"lines": [
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"spans": [
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"bbox": [
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"type": "interline_equation",
"content": "\\log \\mathcal {N} _ {\\llbracket} (\\delta , \\mathcal {P}, d _ {H}) \\leq \\log \\mathcal {N} \\left(c \\sigma_ {\\min } ^ {D + 3} \\delta^ {4}, \\mathcal {F}, \\| | \\cdot | _ {\\infty} \\| _ {\\infty}\\right) + \\log \\left(\\frac {C}{\\sigma_ {\\min } ^ {D + 2} \\delta^ {4}}\\right), \\tag {6}",
"image_path": "d49bfb55d7b0dc776e9377c3c54aeebf97fc3d2206291781f413c93cb1395cca.jpg"
}
]
}
],
"index": 15
},
{
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],
"type": "text",
"angle": 0,
"lines": [
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"bbox": [
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"spans": [
{
"bbox": [
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"type": "text",
"content": "The proof of Lemma 1 is provided in the Appendix E. Theorem 1 presents the convergence rate of the sieve-MLE to the true distribution (see Appendix F for the proof)."
}
]
}
],
"index": 16
},
{
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],
"type": "text",
"angle": 0,
"lines": [
{
"bbox": [
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],
"spans": [
{
"bbox": [
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542,
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],
"type": "text",
"content": "Theorem 1. Let "
},
{
"bbox": [
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542,
512
],
"type": "inline_equation",
"content": "\\mathcal{F},\\mathcal{P},\\sigma_{\\mathrm{min}}"
},
{
"bbox": [
304,
416,
542,
512
],
"type": "text",
"content": " and "
},
{
"bbox": [
304,
416,
542,
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],
"type": "inline_equation",
"content": "\\delta_{*} = \\delta_{*}(D)"
},
{
"bbox": [
304,
416,
542,
512
],
"type": "text",
"content": " be given as in Lemma 1, and "
},
{
"bbox": [
304,
416,
542,
512
],
"type": "inline_equation",
"content": "n\\geq 1"
},
{
"bbox": [
304,
416,
542,
512
],
"type": "text",
"content": ". Suppose that "
},
{
"bbox": [
304,
416,
542,
512
],
"type": "inline_equation",
"content": "\\log \\mathcal{N}(\\delta ,\\mathcal{F},\\| |\\cdot |_{\\infty}\\|_{\\infty})\\leq \\xi \\left\\{A + 1\\vee \\log \\delta^{-1}\\right\\}"
},
{
"bbox": [
304,
416,
542,
512
],
"type": "text",
"content": " for every "
},
{
"bbox": [
304,
416,
542,
512
],
"type": "inline_equation",
"content": "\\delta \\in (0,\\delta_{*}]"
},
{
"bbox": [
304,
416,
542,
512
],
"type": "text",
"content": " and some "
},
{
"bbox": [
304,
416,
542,
512
],
"type": "inline_equation",
"content": "A,\\xi >0"
},
{
"bbox": [
304,
416,
542,
512
],
"type": "text",
"content": ". Suppose that there exists a "
},
{
"bbox": [
304,
416,
542,
512
],
"type": "inline_equation",
"content": "G\\in \\mathcal{F}"
},
{
"bbox": [
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416,
542,
512
],
"type": "text",
"content": " and some "
},
{
"bbox": [
304,
416,
542,
512
],
"type": "inline_equation",
"content": "\\delta_{\\mathrm{approx}}\\in (0,\\delta_{*}]"
},
{
"bbox": [
304,
416,
542,
512
],
"type": "text",
"content": " such that "
},
{
"bbox": [
304,
416,
542,
512
],
"type": "inline_equation",
"content": "\\| \\| G - G_{*}|_{\\infty}\\|_{\\infty}\\leq \\delta_{\\mathrm{approx}}"
},
{
"bbox": [
304,
416,
542,
512
],
"type": "text",
"content": ". Furthermore, suppose that "
},
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"type": "text",
"content": " and "
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"type": "text",
"content": ", and gain insight into their approximation capabilities for the composite structure class described in Assumption 2."
}
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}
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"type": "text",
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"content": "For the sparse neural architecture class "
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"content": "\\mathcal{F}_s(L,r,s,K)"
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"type": "text",
"content": ", the entropy, formally stated as Proposition 1 in Ohn & Kim (2019), is bounded as follows."
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"content": "\\log \\mathcal {N} (\\delta , \\mathcal {F} _ {s}, \\| | \\cdot | _ {\\infty} \\| _ {\\infty}) \\lesssim s L \\left\\{\\log (B L r) + \\log \\delta^ {- 1} \\right\\}. \\tag {10}",
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"type": "text",
"content": " can be viewed as "
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"content": " without any sparsity constraint, meaning "
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"content": "s\\asymp r^2 L"
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"type": "text",
"content": ". Therefore, we have"
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"type": "interline_equation",
"content": "\\log \\mathcal {N} (\\delta , \\mathcal {F} _ {c}, \\| | \\cdot | _ {\\infty} \\| _ {\\infty}) \\lesssim L ^ {2} r ^ {2} \\left\\{\\log (B L r) + \\log \\delta^ {- 1} \\right\\}. \\tag {11}",
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"type": "text",
"content": "The approximation properties of the sparse and fully connected network are provided in Lemma 4.1 and Lemma 4.2 of the Appendix K, respectively."
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"content": " in (10) and Lemma 4.1, respectively, we can simplify Theorem 1 and state Corollary 1."
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"type": "text",
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"content": "\\sigma_{\\max} < \\infty"
},
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"content": ". Moreover, assume that the noise "
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},
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"content": " decays at rate "
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"content": ", i.e., "
},
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},
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"type": "text",
"content": ", and "
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"content": "\\sigma_{\\min} = n^{-\\gamma}"
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"type": "text",
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},
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"content": "\\delta_{\\mathrm{approx}} \\in [0,1]"
},
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"type": "text",
"content": ", the following holds:"
}
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"type": "text",
"content": "1. Let "
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"content": "\\mathcal{F}_s = \\mathcal{F}_s(L,r,s,B,K)"
},
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"type": "text",
"content": " with "
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"content": "\\delta_{*} = \\delta_{*}(D)"
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"type": "text",
"content": " be as given in Lemma 1, and "
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"content": "L\\asymp \\log \\delta_{\\mathrm{approx}}^{-1}"
},
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"type": "text",
"content": ", "
},
{
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"content": "r\\asymp \\delta_{\\mathrm{approx}}^{-t_{*} / \\beta_{*}}"
},
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"type": "text",
"content": ", "
},
{
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],
"type": "inline_equation",
"content": "s\\asymp \\delta_{\\mathrm{approx}}^{-t_{*} / \\beta_{*}}\\log \\delta_{\\mathrm{approx}}^{-1}"
},
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"bbox": [
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],
"type": "text",
"content": ", "
},
{
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234,
543,
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],
"type": "inline_equation",
"content": "B\\asymp \\delta_{\\mathrm{approx}}^{-1}"
},
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],
"type": "text",
"content": ". Then the sieve MLE "
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"content": "\\widehat{p}_{s}"
},
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"type": "text",
"content": " satisfies (7) with "
},
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],
"type": "inline_equation",
"content": "\\varepsilon_n^*"
},
{
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"type": "text",
"content": " as in (8) with "
},
{
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],
"type": "inline_equation",
"content": "\\xi = \\delta_{\\mathrm{approx}}^{-t_{*} / \\beta_{*}}\\log^{2}(\\delta_{\\mathrm{approx}}^{-1})"
},
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"type": "text",
"content": " and "
},
{
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"type": "inline_equation",
"content": "A = \\log^{2}(\\delta_{\\mathrm{approx}}^{-1})"
},
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"type": "text",
"content": " provided that "
},
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"content": "\\eta_n\\leq n\\varepsilon_n^{*2} / 6"
},
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"type": "text",
"content": " and "
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"content": "\\varepsilon_n^*\\leq \\sqrt{2}\\delta_*"
},
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"type": "text",
"content": "."
}
]
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"type": "text",
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"content": "2. Let "
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"content": "\\mathcal{F}_c = \\mathcal{F}_c(L,r,B,K)"
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"type": "text",
"content": " with "
},
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"type": "inline_equation",
"content": "\\delta_{*} = \\delta_{*}(D)"
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"content": " be as given in Lemma 1, and "
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"type": "inline_equation",
"content": "L\\asymp \\log \\delta_{\\mathrm{approx}}^{-1}"
},
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"type": "text",
"content": ", "
},
{
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],
"type": "inline_equation",
"content": "r\\asymp \\delta_{\\mathrm{approx}}^{-t_* / 2\\beta_*}"
},
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"type": "text",
"content": ", "
},
{
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"content": "B\\asymp \\delta_{\\mathrm{approx}}^{-1}"
},
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"type": "text",
"content": ". Then the sieve MLE "
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],
"type": "inline_equation",
"content": "\\widehat{p}_c"
},
{
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],
"type": "text",
"content": " satisfies (7) with "
},
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312,
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"type": "inline_equation",
"content": "\\varepsilon_n^*"
},
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"content": " as in (8) with "
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"content": "\\xi = \\delta_{\\mathrm{approx}}^{-t_* / \\beta_*}\\log^2 (\\delta_{\\mathrm{approx}}^{-1})"
},
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"content": " and "
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"content": "A = \\log^{2}(\\delta_{\\mathrm{approx}}^{-1})"
},
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"type": "text",
"content": " provided that "
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},
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"type": "text",
"content": " and "
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"content": "\\varepsilon_{n}^{*}\\leq \\sqrt{2}\\delta_{*}"
},
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"type": "text",
"content": "."
}
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"type": "text",
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"content": "\\delta_{\\mathrm{approx}} \\coloneqq \\left( \\sigma_{*}^{2} / n \\right)^{\\beta_{*} / (2\\beta_{*} + t_{*})}"
},
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"type": "text",
"content": " minimizes "
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"content": "\\varepsilon_{n}^{*} \\asymp \\sqrt{\\xi \\left\\{ A + \\log \\left( n / \\sigma_{\\min} \\right) \\right\\} / n} \\vee \\delta_{\\mathrm{approx}} / \\sigma_{*}"
},
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"type": "text",
"content": ", and gives"
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"content": "\\varepsilon_ {n} ^ {*} \\asymp n ^ {- \\frac {\\beta_ {*} - t _ {*} \\alpha}{2 \\beta_ {*} + t _ {*}}} \\log^ {2} (n). \\tag {12}",
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"type": "text",
"content": " is upper bounded as "
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"content": "\\varepsilon_{n}^{*}\\leq \\sqrt{2}\\delta_{*}(D)"
},
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"type": "text",
"content": ". For large values of "
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"type": "inline_equation",
"content": "\\alpha"
},
{
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"type": "text",
"content": ", estimation of "
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"type": "inline_equation",
"content": "G_{*}"
},
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"type": "text",
"content": " is inherent difficult as the data is very close on the singular support. To address this, a small noise injection, as described in Corollary 2, can smooth the estimation and ensure consistency."
}
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}
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"type": "text",
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"content": "The proof of Corollary 1 is provided in Appendix G. For the composite structural class "
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"type": "inline_equation",
"content": "\\mathcal{G}"
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"type": "text",
"content": ", the effective smoothness is denoted by "
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"content": "\\beta_{*}"
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"type": "text",
"content": ", and the dimension is "
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"content": "t_*"
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"type": "text",
"content": ". This effectively mitigates the curse of dimensionality. The convergence rate at (12) also recovers the optimal rate when "
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"type": "inline_equation",
"content": "q = 1"
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"type": "text",
"content": " and "
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"type": "text",
"content": ", and there is a small lag of polynomial factor "
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"content": "t_*\\alpha / (2\\beta_* + t_*)"
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"type": "text",
"content": " when "
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"type": "inline_equation",
"content": "\\alpha > 0"
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"content": ". See Appendix B for their formal definitions."
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"content": " for some "
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"content": "\\beta_Z, \\beta_X, \\beta_Q > 0"
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"content": " and are bounded above and below. Then, there exists a map "
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"type": "text",
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"content": ". A push-forward map is a transport map between two distributions. The well-established regularity theory of transport map in optimal transport is directly applicable here [see Villani et al. (2009) and Villani (2021)]. The proof of Lemma 2 is based on Theorem 12.50 of (Villani et al., 2009) and Caffarelli (1996), which establishes the regularity of this transport map and its existence follows from Brenier (1991). When "
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"content": ". In particular, choosing "
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"content": "\\delta_{\\mathrm{approx}}:= (\\sigma_{*}^{2} / n)^{1 / (2 + \\beta_{Z}^{-1}\\mathfrak{d} + \\beta_{X}^{-1}\\mathfrak{p})}\\leq 1"
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"content": " minimizes "
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"content": "\\varepsilon_n^*\\asymp \\sqrt{\\xi\\left\\{A + \\log\\left(n / \\sigma_{\\mathrm{min}}\\right)\\right\\} / n}\\vee \\delta_{\\mathrm{approx}} / \\sigma_*"
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"content": "\\varepsilon_ {n} ^ {*} \\asymp n ^ {- \\frac {1 - \\alpha \\left(\\beta_ {Z} ^ {- 1} \\mathfrak {d} + \\beta_ {X} ^ {- 1} \\mathfrak {p}\\right)}{2 + \\beta_ {Z} ^ {- 1} \\mathfrak {d} + \\beta_ {X} ^ {- 1} \\mathfrak {p}}} \\log^ {2} (n). \\tag {14}",
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"content": ", our convergence rate in (14) recovers the minimax optimal rate for conditional density estimation based on kernel smoothing, as established in (Li et al., 2022)."
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"content": "We compare the sieve MLE with CKDE (Hall et al., 2004) and FlexCode proposed by Izbicki & Lee (2017). To evaluate their performance, we compute the mean squared error (MSE) for both the mean and the standard deviation. We use Monte Carlo approximation to compute the mean and standard deviation for the sieve MLE, and numerical integration for CKDE and Flexcode. This evaluation strategy resembles that implemented by Zhou et al. (2022). Table 2 summarizes the findings."
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"html": " | | Sieve MLE | CKDE | FlexCode |
| FD1 | MEAN | 0.0379 ± 0.0170 | 1.0053 ± 0.1004 | 1.1660 ± 0.1076 |
| SD | 0.0280 ± 0.0045 | 0.9887 ± 0.0347 | 1.2000 ± 0.0126 |
| FD2 | MEAN | 0.1943 ± 0.0427 | 0.2640 ± 0.0515 | 0.3954 ± 0.0571 |
| SD | 0.2843 ± 0.0093 | 0.2853 ± 0.0213 | 5.8278 ± 0.1607 |
| FD3 | MEAN | 0.2337 ± 0.0453 | 0.2967 ± 0.0537 | 1.3419 ± 0.1087 |
| SD | 1.6394 ± 0.0861 | 0.6334 ± 0.0460 | 11.4898 ± 0.1559 |
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"html": "| Noise | Sparsely connected | Fully connected |
| 0 | 1.9555 ± 0.7182 | 1.8859 ± 0.7355 |
| 0.005 | 1.9478 ± 0.7329 | 1.8663 ± 0.6251 |
| 0.01 | 1.9503 ± 0.7291 | 1.9598 ± 0.6867 |
| 0.02 | 2.0699 ± 0.6937 | 2.0616 ± 0.7410 |
| 0.04 | 2.2199 ± 0.6735 | 2.2117 ± 0.6627 |
| 0.06 | 2.3487 ± 0.6576 | 2.3172 ± 0.6267 |
| 0.08 | 2.4623 ± 0.6245 | 2.4076 ± 0.6308 |
| 0.1 | 2.5734 ± 0.6492 | 2.5002 ± 0.6337 |
| 0.3 | 3.4931 ± 0.7012 | 3.4943 ± 0.7164 |
| 0.5 | 4.0880 ± 0.7518 | 4.0995 ± 0.7633 |
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"html": "| Sample Size | Two Moon (σ* = 0.01) | Ellipse (σ* = 0.01) |
| 4000 | 0.251 | 0.295 |
| 6000 | 0.232 | 0.285 |
| 7000 | 0.216 | 0.271 |
| 8000 | 0.214 | 0.253 |
| 9000 | 0.212 | 0.259 |
| 10000 | 0.196 | 0.251 |
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"content": "\\mathcal{H}^{\\beta_{\\mathrm{min}}}(\\mathcal{B}_1(0_{\\mathrm{d}_*}))"
},
{
"bbox": [
52,
163,
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],
"type": "text",
"content": ". By Lemma 2, assuming "
},
{
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52,
163,
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213
],
"type": "inline_equation",
"content": "\\beta_{\\mathrm{min}} \\leq \\beta_Z \\wedge \\beta_X"
},
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52,
163,
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213
],
"type": "text",
"content": ", there exists "
},
{
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52,
163,
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213
],
"type": "inline_equation",
"content": "g \\in \\mathcal{H}^{\\beta_{\\mathrm{min}} + 1}"
},
{
"bbox": [
52,
163,
544,
213
],
"type": "text",
"content": " such that "
},
{
"bbox": [
52,
163,
544,
213
],
"type": "inline_equation",
"content": "\\widetilde{Q} = Q_g"
},
{
"bbox": [
52,
163,
544,
213
],
"type": "text",
"content": ". Thus, we have "
},
{
"bbox": [
52,
163,
544,
213
],
"type": "inline_equation",
"content": "Q = Q_{\\varphi \\circ g}"
},
{
"bbox": [
52,
163,
544,
213
],
"type": "text",
"content": ", where "
},
{
"bbox": [
52,
163,
544,
213
],
"type": "inline_equation",
"content": "\\varphi \\circ g: \\mathcal{Z} \\times \\mathcal{X} \\to \\mathcal{Y}"
},
{
"bbox": [
52,
163,
544,
213
],
"type": "text",
"content": ". Following Lemma 4, it is possible to find the appropriate neural network approximating them."
}
]
}
],
"index": 4
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{
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217,
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"type": "text",
"angle": 0,
"lines": [
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"spans": [
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],
"type": "text",
"content": "Suppose "
},
{
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217,
544,
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],
"type": "inline_equation",
"content": "\\mathcal{V}"
},
{
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52,
217,
544,
266
],
"type": "text",
"content": " is covered by the charts "
},
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"bbox": [
52,
217,
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266
],
"type": "inline_equation",
"content": "\\{(U_k,\\varphi_k)\\}_{k = 1}^K"
},
{
"bbox": [
52,
217,
544,
266
],
"type": "text",
"content": ", with "
},
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"bbox": [
52,
217,
544,
266
],
"type": "inline_equation",
"content": "1 < K < \\infty"
},
{
"bbox": [
52,
217,
544,
266
],
"type": "text",
"content": ", where "
},
{
"bbox": [
52,
217,
544,
266
],
"type": "inline_equation",
"content": "\\varphi_{k}:\\mathcal{B}_{1}(0_{\\mathrm{d}_{*}})\\to U_{k}"
},
{
"bbox": [
52,
217,
544,
266
],
"type": "text",
"content": " is a homeomorphism. As before, we assume "
},
{
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52,
217,
544,
266
],
"type": "inline_equation",
"content": "\\varphi_{k}\\in \\mathcal{H}^{\\beta_{\\min} + 1}"
},
{
"bbox": [
52,
217,
544,
266
],
"type": "text",
"content": ", "
},
{
"bbox": [
52,
217,
544,
266
],
"type": "inline_equation",
"content": "|J_{\\varphi_k}(\\mathbf{u})|"
},
{
"bbox": [
52,
217,
544,
266
],
"type": "text",
"content": " is bounded below by a positive constant, "
},
{
"bbox": [
52,
217,
544,
266
],
"type": "inline_equation",
"content": "Q"
},
{
"bbox": [
52,
217,
544,
266
],
"type": "text",
"content": " possesses density "
},
{
"bbox": [
52,
217,
544,
266
],
"type": "inline_equation",
"content": "q"
},
{
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217,
544,
266
],
"type": "text",
"content": " with respect to "
},
{
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266
],
"type": "inline_equation",
"content": "\\mathrm{H_{d_*}}"
},
{
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217,
544,
266
],
"type": "text",
"content": " that is bounded above and below, and that "
},
{
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217,
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266
],
"type": "inline_equation",
"content": "q\\circ \\phi_{k}\\in \\mathcal{H}^{\\beta_{\\min}}"
},
{
"bbox": [
52,
217,
544,
266
],
"type": "text",
"content": ". Let "
},
{
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52,
217,
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266
],
"type": "inline_equation",
"content": "Q_{k}(\\cdot) = Q(\\cdot) / Q(U_{k})"
},
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217,
544,
266
],
"type": "text",
"content": " be the normalized measure of "
},
{
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52,
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544,
266
],
"type": "inline_equation",
"content": "Q"
},
{
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52,
217,
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266
],
"type": "text",
"content": " over "
},
{
"bbox": [
52,
217,
544,
266
],
"type": "inline_equation",
"content": "U_{k}"
},
{
"bbox": [
52,
217,
544,
266
],
"type": "text",
"content": "."
}
]
}
],
"index": 5
},
{
"bbox": [
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272,
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],
"type": "text",
"angle": 0,
"lines": [
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272,
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"spans": [
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],
"type": "text",
"content": "We denote "
},
{
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272,
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319
],
"type": "inline_equation",
"content": "q_{k}"
},
{
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52,
272,
544,
319
],
"type": "text",
"content": " as the corresponding density with respect to "
},
{
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52,
272,
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319
],
"type": "inline_equation",
"content": "\\mathsf{H}_{\\mathbf{d}_{*}}"
},
{
"bbox": [
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272,
544,
319
],
"type": "text",
"content": ". For "
},
{
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52,
272,
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319
],
"type": "inline_equation",
"content": "\\mathbf{u} \\in U_k \\cap U_\\ell"
},
{
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272,
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319
],
"type": "text",
"content": ", "
},
{
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],
"type": "inline_equation",
"content": "q_{k}(\\mathbf{u})Q(U_{k}) = q_{\\ell}(\\mathbf{u})Q(U_{\\ell}) = q(\\mathbf{u})"
},
{
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52,
272,
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319
],
"type": "text",
"content": " holds due to the measure "
},
{
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52,
272,
544,
319
],
"type": "inline_equation",
"content": "Q(\\cdot)"
},
{
"bbox": [
52,
272,
544,
319
],
"type": "text",
"content": " being compatible with the charts. This is ensured because the densities "
},
{
"bbox": [
52,
272,
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319
],
"type": "inline_equation",
"content": "Q(U_{k})q_{k}(\\cdot)"
},
{
"bbox": [
52,
272,
544,
319
],
"type": "text",
"content": " and "
},
{
"bbox": [
52,
272,
544,
319
],
"type": "inline_equation",
"content": "Q(U_{\\ell})q_{\\ell}(\\cdot)"
},
{
"bbox": [
52,
272,
544,
319
],
"type": "text",
"content": " are consistent and align with the measure "
},
{
"bbox": [
52,
272,
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319
],
"type": "inline_equation",
"content": "Q"
},
{
"bbox": [
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272,
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319
],
"type": "text",
"content": " over the overlapping regions of the charts. This compatibility is essential for constructing a coherent global measure from local chart densities."
}
]
}
],
"index": 6
},
{
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],
"type": "text",
"angle": 0,
"lines": [
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],
"type": "text",
"content": "A compact manifold "
},
{
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326,
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],
"type": "inline_equation",
"content": "\\mathcal{V}"
},
{
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52,
326,
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],
"type": "text",
"content": " can be covered by a finite partition of unity "
},
{
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],
"type": "inline_equation",
"content": "\\{\\tau_k, k = 1, \\dots, K\\}"
},
{
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326,
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],
"type": "text",
"content": ", each sufficiently smooth (Lee, 2012). By definition, each function in this partition satisfies "
},
{
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52,
326,
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],
"type": "inline_equation",
"content": "\\tau_k(\\mathbf{u}) = 0"
},
{
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326,
544,
364
],
"type": "text",
"content": " for "
},
{
"bbox": [
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326,
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],
"type": "inline_equation",
"content": "\\mathbf{u} \\notin U_k"
},
{
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364
],
"type": "text",
"content": " and "
},
{
"bbox": [
52,
326,
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],
"type": "inline_equation",
"content": "\\sum_{k=1}^{K} \\tau_k(\\mathbf{u}) = 1"
},
{
"bbox": [
52,
326,
544,
364
],
"type": "text",
"content": " for all "
},
{
"bbox": [
52,
326,
544,
364
],
"type": "inline_equation",
"content": "\\mathbf{u} \\in \\mathcal{V}"
},
{
"bbox": [
52,
326,
544,
364
],
"type": "text",
"content": ". Given that "
},
{
"bbox": [
52,
326,
544,
364
],
"type": "inline_equation",
"content": "q(\\mathbf{u}) = Q(U_k) q_k(\\mathbf{u})"
},
{
"bbox": [
52,
326,
544,
364
],
"type": "text",
"content": " for each "
},
{
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52,
326,
544,
364
],
"type": "inline_equation",
"content": "k"
},
{
"bbox": [
52,
326,
544,
364
],
"type": "text",
"content": " and "
},
{
"bbox": [
52,
326,
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364
],
"type": "inline_equation",
"content": "\\mathbf{u} \\in U_k"
},
{
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52,
326,
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364
],
"type": "text",
"content": ", we can express "
},
{
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326,
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],
"type": "inline_equation",
"content": "q(\\mathbf{u})"
},
{
"bbox": [
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],
"type": "text",
"content": " as:"
}
]
}
],
"index": 7
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{
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231,
369,
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],
"type": "interline_equation",
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"spans": [
{
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],
"type": "interline_equation",
"content": "q (\\mathbf {u}) = \\sum_ {k = 1} ^ {K} Q (U _ {k}) \\tau_ {k} (\\mathbf {u}) q _ {k} (\\mathbf {u}).",
"image_path": "0625a0e9a0522a77b0ea8ae8538f1793f1d87e608e9bfe50a921619839926c93.jpg"
}
]
}
],
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"type": "text",
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],
"type": "text",
"content": "To normalize, let "
},
{
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409,
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],
"type": "inline_equation",
"content": "c_k = \\int \\tau_k(\\mathbf{u}) dQ_k(\\mathbf{u})"
},
{
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52,
409,
480,
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],
"type": "text",
"content": " and define "
},
{
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409,
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],
"type": "inline_equation",
"content": "q_k'(\\mathbf{u}) = \\tau_k(\\mathbf{u}) q_k(\\mathbf{u}) / c_k"
},
{
"bbox": [
52,
409,
480,
422
],
"type": "text",
"content": ". Thus, we can rewrite "
},
{
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52,
409,
480,
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],
"type": "inline_equation",
"content": "q(\\mathbf{u})"
},
{
"bbox": [
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],
"type": "text",
"content": " as:"
}
]
}
],
"index": 9
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"type": "interline_equation",
"angle": 0,
"lines": [
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"spans": [
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],
"type": "interline_equation",
"content": "q (\\mathbf {u}) = \\sum_ {k = 1} ^ {K} \\pi_ {k} q _ {k} ^ {\\prime} (\\mathbf {u}),",
"image_path": "0f5985cf37e4f8adbf663017448f49a7544ad2ce754c085d2a83d2341cf65a6e.jpg"
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]
}
],
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{
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],
"type": "text",
"angle": 0,
"lines": [
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],
"type": "text",
"content": "where "
},
{
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52,
468,
544,
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],
"type": "inline_equation",
"content": "\\pi_k = c_k Q(U_k)"
},
{
"bbox": [
52,
468,
544,
503
],
"type": "text",
"content": ". This formulation reveals that "
},
{
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52,
468,
544,
503
],
"type": "inline_equation",
"content": "q"
},
{
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468,
544,
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],
"type": "text",
"content": " is a mixture of the component densities "
},
{
"bbox": [
52,
468,
544,
503
],
"type": "inline_equation",
"content": "q_k'(\\mathbf{u})"
},
{
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468,
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],
"type": "text",
"content": ", weighted by "
},
{
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],
"type": "inline_equation",
"content": "\\pi_k"
},
{
"bbox": [
52,
468,
544,
503
],
"type": "text",
"content": ". This mixture approach ensures compatibility across different charts, providing a unified density representation over the entire manifold "
},
{
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468,
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],
"type": "inline_equation",
"content": "\\mathcal{Y}"
},
{
"bbox": [
52,
468,
544,
503
],
"type": "text",
"content": "."
}
]
}
],
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{
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],
"type": "text",
"angle": 0,
"lines": [
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],
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],
"type": "text",
"content": "Since "
},
{
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544,
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],
"type": "inline_equation",
"content": "q_{k}^{\\prime}"
},
{
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544,
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],
"type": "text",
"content": " is sufficiently smooth, we can construct a mapping "
},
{
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],
"type": "inline_equation",
"content": "g_{k}: \\widetilde{\\mathcal{V}} \\to \\mathcal{V}"
},
{
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510,
544,
611
],
"type": "text",
"content": " such that "
},
{
"bbox": [
52,
510,
544,
611
],
"type": "inline_equation",
"content": "Q_{k}^{\\prime}"
},
{
"bbox": [
52,
510,
544,
611
],
"type": "text",
"content": " is the distribution of "
},
{
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510,
544,
611
],
"type": "inline_equation",
"content": "g_{k}(\\widetilde{V})"
},
{
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510,
544,
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],
"type": "text",
"content": ", supported on "
},
{
"bbox": [
52,
510,
544,
611
],
"type": "inline_equation",
"content": "U_{k}"
},
{
"bbox": [
52,
510,
544,
611
],
"type": "text",
"content": ", where "
},
{
"bbox": [
52,
510,
544,
611
],
"type": "inline_equation",
"content": "\\widetilde{\\mathcal{V}}"
},
{
"bbox": [
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510,
544,
611
],
"type": "text",
"content": " is a uniformly convex set in "
},
{
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],
"type": "inline_equation",
"content": "\\mathbb{R}^{d_{*}}"
},
{
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],
"type": "text",
"content": ", and "
},
{
"bbox": [
52,
510,
544,
611
],
"type": "inline_equation",
"content": "\\widetilde{V}"
},
{
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510,
544,
611
],
"type": "text",
"content": " follows a uniform distribution on "
},
{
"bbox": [
52,
510,
544,
611
],
"type": "inline_equation",
"content": "\\widetilde{\\mathcal{V}}"
},
{
"bbox": [
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510,
544,
611
],
"type": "text",
"content": ". Next, construct a disjoint partition of the interval "
},
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],
"type": "inline_equation",
"content": "(0,1)"
},
{
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],
"type": "text",
"content": " into "
},
{
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],
"type": "inline_equation",
"content": "K"
},
{
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],
"type": "text",
"content": " intervals "
},
{
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],
"type": "inline_equation",
"content": "I_{1}, \\ldots, I_{K}"
},
{
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],
"type": "text",
"content": " with lengths "
},
{
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],
"type": "inline_equation",
"content": "\\pi_{1}, \\ldots, \\pi_{K}"
},
{
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],
"type": "text",
"content": ", where "
},
{
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],
"type": "inline_equation",
"content": "I_{k} = [\\sum_{i=1}^{k-1} \\pi_{i}, \\sum_{i=1}^{k} \\pi_{i}]"
},
{
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611
],
"type": "text",
"content": ". Define "
},
{
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544,
611
],
"type": "inline_equation",
"content": "h_{k}"
},
{
"bbox": [
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],
"type": "text",
"content": " as the indicator function on the interval "
},
{
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],
"type": "inline_equation",
"content": "I_{k}"
},
{
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611
],
"type": "text",
"content": ", i.e., "
},
{
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510,
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611
],
"type": "inline_equation",
"content": "h_{k}(u) = 1"
},
{
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],
"type": "text",
"content": " if "
},
{
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"content": "P_{\\mathsf{U}}(h_{k}(\\mathsf{U}) = 1) = \\pi_{k}"
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"content": "P_{\\mathsf{U}}(h_{k}(\\mathsf{U}) = 0) = 1 - \\pi_{k}"
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"content": ". Now, define "
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"content": "\\mathbf{v} = (\\mathsf{u},\\widetilde{v})"
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"content": ". Using this, construct "
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"content": "g(\\mathbf{v}) = \\sum_{k=1}^{K} h_{k}(\\mathbf{u}) g_{k}(v)"
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"content": ". It is straightforward to observe that "
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"content": "Q = Q_{g}"
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"content": ", as the partitioning through "
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"content": " ensures that the measure is correctly matched to each "
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"content": " are appropriately supported on "
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"content": "."
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"content": "From an approximation perspective, the indicator functions "
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"content": " and the localized generators can be effectively approximated using ReLU neural networks. This also holds for their products and further linear combinations. For details on such constructions, one may refer to Schmidt-Hieber (2019) for sparse neural networks and Kohler et al. (2023) for dense neural networks."
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"content": " maps, as they are not necessarily lower bounded. However, the partition of unity maps "
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"content": "\\tau_{k}"
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"content": " vanish only at the boundary of "
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"content": ". This property may allows for the construction of sufficiently smooth maps. For the multiple-chart case, we rely on more stringent results, such as Brenier's Theorem (see, for example, Villani et al. (2009)) or the Noise Outsourcing Lemma (Lemma 3), to ensure the existence of the transport maps."
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"content": "| v | _ {1} \\phi_ {\\sigma} (v) \\leq \\sqrt {\\frac {D}{2 \\pi e}} \\frac {1}{(\\sqrt {2 \\pi \\sigma^ {2}}) ^ {D - 1}}.",
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"content": "\\begin{array}{l} p _ {g, \\sigma_ {1}} (y | x) - p _ {g _ {2}, \\sigma_ {2}} (y | x) \\\\ = \\int \\phi_ {\\sigma_ {1}} (y - g (x, z) \\left(1 - \\left(\\frac {\\sigma_ {1}}{\\sigma_ {2}}\\right) ^ {D} \\exp \\left\\{\\frac {| y - g (x , z) | _ {2} ^ {2}}{2} \\left(\\frac {1}{\\sigma_ {1} ^ {2}} - \\frac {1}{\\sigma_ {2} ^ {2}}\\right) \\right\\}\\right) d P _ {Z} (z) \\\\ \\leq \\int \\phi_ {\\sigma_ {1}} (y - g (x, z) \\left[ \\frac {| y - g (x , z) | _ {2} ^ {2}}{2} \\left(\\frac {1}{\\sigma_ {2} ^ {2}} - \\frac {1}{\\sigma_ {1} ^ {2}}\\right) - D \\log \\left(\\frac {\\sigma_ {1}}{\\sigma_ {2}}\\right) \\right] d P _ {Z} (z) (22) \\\\ \\leq \\int \\phi_ {\\sigma_ {1}} (y - g (x, z) \\left[ \\frac {| y - g (x , z) | _ {2} ^ {2}}{2} \\left(\\frac {\\sigma_ {1} + \\sigma_ {2}}{\\sigma_ {1} ^ {2} \\sigma_ {2} ^ {2}}\\right) \\eta_ {2} + \\frac {D \\eta_ {2}}{\\min \\{\\sigma_ {1} , \\sigma_ {2} \\}} \\right] d P _ {Z} (z) \\\\ \\leq \\frac {1}{\\left(\\sqrt {2 \\pi \\sigma_ {1} ^ {2}}\\right) ^ {D}} \\frac {\\sigma_ {1} + \\sigma_ {2}}{e \\sigma_ {2} ^ {2}} \\eta_ {2} + \\frac {1}{\\left(\\sqrt {2 \\pi \\sigma_ {1} ^ {2}}\\right) ^ {D}} \\frac {D \\eta_ {2}}{\\min \\left\\{\\sigma_ {1} , \\sigma_ {2} \\right\\}} (23) \\\\ \\leq c _ {2} (D) \\sigma_ {\\min } ^ {- (D + 1)} \\eta_ {2}. (24) \\\\ \\end{array}",
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"content": "P _ {*} \\left(d _ {H} (\\widehat {p}, p _ {*}) > \\varepsilon_ {n}\\right) \\leq 5 e ^ {- c _ {2} n \\varepsilon_ {n} ^ {* 2}} + \\frac {\\tau_ {n}}{n \\delta_ {n}} = 5 e ^ {- c _ {2} n \\varepsilon_ {n} ^ {* 2}} + \\frac {2 c _ {7} ^ {2}}{D n}.",
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"type": "text",
"content": "H. Proof of Theorem 2"
}
]
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{
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"type": "text",
"angle": 0,
"lines": [
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"type": "text",
"content": "Proof. It is suffice to assume that "
},
{
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],
"type": "inline_equation",
"content": "\\varepsilon"
},
{
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52,
238,
542,
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],
"type": "text",
"content": " and "
},
{
"bbox": [
52,
238,
542,
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],
"type": "inline_equation",
"content": "\\sigma_{*}\\sqrt{\\log\\varepsilon^{-1}}"
},
{
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52,
238,
542,
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],
"type": "text",
"content": " are sufficiently small. If not, let "
},
{
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238,
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],
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"content": "\\varepsilon +\\sigma_{*}\\sqrt{\\log\\varepsilon^{-1}}\\geq c_{0}"
},
{
"bbox": [
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],
"type": "text",
"content": ", where "
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{
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"type": "inline_equation",
"content": "c_{0}(K,D,\\mathsf{r}_{*})"
},
{
"bbox": [
52,
238,
542,
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],
"type": "text",
"content": ". Then Theorem 2 holds trivially by taking a large enough constant depending just on "
},
{
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52,
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],
"type": "inline_equation",
"content": "D,K"
},
{
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],
"type": "text",
"content": ", and "
},
{
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],
"type": "inline_equation",
"content": "\\mathsf{r}_*"
}
]
}
],
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"type": "text",
"angle": 0,
"lines": [
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"spans": [
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"type": "text",
"content": "Let "
},
{
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267,
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],
"type": "inline_equation",
"content": "V \\sim Q(\\cdot |X = x)"
},
{
"bbox": [
52,
267,
543,
342
],
"type": "text",
"content": ", "
},
{
"bbox": [
52,
267,
543,
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],
"type": "inline_equation",
"content": "V_{*} \\sim Q(\\cdot |X = x)"
},
{
"bbox": [
52,
267,
543,
342
],
"type": "text",
"content": ", "
},
{
"bbox": [
52,
267,
543,
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],
"type": "inline_equation",
"content": "\\epsilon \\sim \\mathsf{N}(0_D, \\sigma^2\\mathbb{I}_d)"
},
{
"bbox": [
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267,
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342
],
"type": "text",
"content": " and "
},
{
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],
"type": "inline_equation",
"content": "\\epsilon_{*} \\sim \\mathsf{N}(0_D, \\sigma_{*}^{2}\\mathbb{I}_{d})"
},
{
"bbox": [
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267,
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342
],
"type": "text",
"content": " be independent with underlying probability density "
},
{
"bbox": [
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342
],
"type": "inline_equation",
"content": "\\nu"
},
{
"bbox": [
52,
267,
543,
342
],
"type": "text",
"content": ". We truncate the random variable "
},
{
"bbox": [
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267,
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342
],
"type": "inline_equation",
"content": "\\epsilon"
},
{
"bbox": [
52,
267,
543,
342
],
"type": "text",
"content": " and "
},
{
"bbox": [
52,
267,
543,
342
],
"type": "inline_equation",
"content": "\\epsilon_{*}"
},
{
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267,
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],
"type": "text",
"content": " componentwise as "
},
{
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267,
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],
"type": "inline_equation",
"content": "(\\epsilon_{K})_{j} = \\max \\{-K, \\min \\{K, \\epsilon_{j}\\}\\}"
},
{
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342
],
"type": "text",
"content": " and "
},
{
"bbox": [
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267,
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],
"type": "inline_equation",
"content": "(\\epsilon_{*K})_j = \\max \\{-K, \\min \\{K, (\\epsilon_{*})_j\\}\\}"
},
{
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267,
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],
"type": "text",
"content": " respectively. We denote "
},
{
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267,
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342
],
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"content": "P_{g,\\sigma}"
},
{
"bbox": [
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],
"type": "text",
"content": " as "
},
{
"bbox": [
52,
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],
"type": "inline_equation",
"content": "P"
},
{
"bbox": [
52,
267,
543,
342
],
"type": "text",
"content": ", "
},
{
"bbox": [
52,
267,
543,
342
],
"type": "inline_equation",
"content": "Q_{g}"
},
{
"bbox": [
52,
267,
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],
"type": "text",
"content": " as "
},
{
"bbox": [
52,
267,
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],
"type": "inline_equation",
"content": "Q"
},
{
"bbox": [
52,
267,
543,
342
],
"type": "text",
"content": ", "
},
{
"bbox": [
52,
267,
543,
342
],
"type": "inline_equation",
"content": "\\widetilde{P}"
},
{
"bbox": [
52,
267,
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342
],
"type": "text",
"content": " as distribution of "
},
{
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],
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"content": "V + \\epsilon_{K}"
},
{
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],
"type": "text",
"content": " and "
},
{
"bbox": [
52,
267,
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],
"type": "inline_equation",
"content": "\\widetilde{P}_{*}"
},
{
"bbox": [
52,
267,
543,
342
],
"type": "text",
"content": " as the distribution of "
},
{
"bbox": [
52,
267,
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342
],
"type": "inline_equation",
"content": "V_{*} + \\epsilon_{*K}"
},
{
"bbox": [
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267,
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342
],
"type": "text",
"content": ". One may note that "
},
{
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],
"type": "inline_equation",
"content": "W_{1}(\\widetilde{P}_{*}, Q_{*}) \\leq W_{2}(\\widetilde{P}_{*}, Q_{*}) \\leq \\sqrt{\\mathbb{E}[|\\epsilon_{*K}|_2^2]} \\leq \\sqrt{\\mathbb{E}[|\\epsilon_{*}|_2^2]} \\leq \\sigma_*\\sqrt{D}"
},
{
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],
"type": "text",
"content": ". Similarly, "
},
{
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],
"type": "inline_equation",
"content": "W_{1}(\\widetilde{P}, Q) \\leq \\sigma\\sqrt{D}"
},
{
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],
"type": "text",
"content": ". The "
},
{
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],
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"content": "\\ell_1"
},
{
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],
"type": "text",
"content": " diameter of "
},
{
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],
"type": "inline_equation",
"content": "[-2K, 2K]^D"
},
{
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],
"type": "text",
"content": ", where the support of "
},
{
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267,
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],
"type": "inline_equation",
"content": "\\widetilde{P}"
},
{
"bbox": [
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342
],
"type": "text",
"content": " and "
},
{
"bbox": [
52,
267,
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],
"type": "inline_equation",
"content": "\\widetilde{P}_{*}"
},
{
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267,
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],
"type": "text",
"content": ", is 4KD. Observe that"
}
]
}
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"type": "interline_equation",
"content": "W _ {1} \\left(\\widetilde {P} _ {*}, \\widetilde {P}\\right) \\leq 4 K D d _ {1} \\left(\\widetilde {P} _ {*}, \\widetilde {P}\\right) \\leq 4 K D d _ {1} (P _ {*}, P) \\leq 8 K D d _ {H} (P _ {*}, P),",
"image_path": "c3fd842194ef821222b9691a977ba5b8af540307ab2667966b5bff5b6b35f8b4.jpg"
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]
}
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"index": 11
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"type": "text",
"angle": 0,
"lines": [
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"type": "text",
"content": "where the first inequality follows from Theorem 4 of Gibbs & Su (2002), the second inequality follows from the fact the distance between two truncated distributions is always lesser than the original distributions and the last inequality follows from "
},
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"type": "inline_equation",
"content": "d_{1} \\leq 2d_{H}"
},
{
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],
"type": "text",
"content": ". Hence,"
}
]
}
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{
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"type": "interline_equation",
"content": "W _ {1} \\left(Q _ {*}, Q\\right) \\leq W _ {2} \\left(Q _ {*}, \\widetilde {P} _ {*}\\right) + W _ {1} \\left(\\widetilde {P} _ {*}, \\widetilde {P}\\right) + W _ {2} \\left(\\widetilde {P}, Q\\right) \\leq \\sigma_ {*} \\sqrt {D} + 8 K D \\varepsilon + \\sigma \\sqrt {D}.",
"image_path": "a6d3d441f2f7a83806f4cf50526cd71bb0e049f482128f7ab6645892f6f1c84b.jpg"
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"lines": [
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"type": "text",
"content": "Now it is suffice to show that "
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"type": "inline_equation",
"content": "\\sigma \\leq c\\sigma_{*}\\sqrt{\\log\\varepsilon^{-1}}"
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{
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"type": "text",
"content": ", where "
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{
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],
"type": "inline_equation",
"content": "c = c(D,K,r*)"
},
{
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],
"type": "text",
"content": " is a constant, because we have assumed that "
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{
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"content": "\\varepsilon"
},
{
"bbox": [
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457,
542,
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],
"type": "text",
"content": " is small enough. We establish this in the rest of the proof. Let "
},
{
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],
"type": "inline_equation",
"content": "t_* = \\left[2\\sigma_*^2 D\\log \\left(\\frac{2D}{\\varepsilon}\\right)\\right]^{1/2}"
},
{
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],
"type": "text",
"content": ". Observe that"
}
]
}
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"index": 14
},
{
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"angle": 0,
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"type": "interline_equation",
"content": "\\int_ {| x | _ {2} > t _ {*}} \\phi_ {\\sigma_ {*}} (x) d x \\leq \\int_ {| x | _ {\\infty} > t _ {*} / \\sqrt {D}} \\phi_ {\\sigma_ {*}} (x) d x \\leq 2 D e ^ {- t _ {*} ^ {2} / 2 D \\sigma^ {2}} \\leq \\varepsilon .",
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}
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"type": "text",
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"type": "text",
"content": "Let "
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{
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"content": "\\mathcal{M}_{*}^{t_{*}} = \\mathcal{M}_{*}\\oplus \\mathcal{B}_{t_{*}}(0_{D})"
},
{
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],
"type": "text",
"content": " . We may write"
}
]
}
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"index": 16
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"type": "interline_equation",
"content": "\\begin{array}{l} 1 - P _ {*} \\left(\\mathcal {M} _ {*} ^ {t _ {*}}\\right) = \\nu \\left(Y _ {*} + \\epsilon_ {*} \\notin \\mathcal {M} _ {*} ^ {t _ {*}}\\right) \\leq \\nu \\left(| \\epsilon_ {*} | _ {2} > t _ {*}\\right) \\\\ \\Rightarrow P \\left(\\mathcal {M} _ {*} ^ {t _ {*}}\\right) \\geq 1 - 2 \\varepsilon , \\tag {30} \\\\ \\end{array}",
"image_path": "e9ae5d1a86afc5baa75b0547ab4ff5c49a452591c72a48e71e583efbbad91f15.jpg"
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"type": "text",
"angle": 0,
"lines": [
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],
"type": "text",
"content": "the implication in the last line follows from "
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],
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"content": "\\sup_B |P(B) - P_*(B)| \\leq d_H(P, P_*) \\leq \\varepsilon"
},
{
"bbox": [
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589,
542,
624
],
"type": "text",
"content": ". For the sake of contradiction, let "
},
{
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],
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"content": "\\sigma \\in [2t_* r^* / 2] \\cup (\\mathfrak{r}_* / 2, \\infty)"
},
{
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],
"type": "text",
"content": " ("
},
{
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],
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"content": "t_*"
},
{
"bbox": [
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],
"type": "text",
"content": " is sufficiently small, from the assumption we made at the beginning of this proof). If "
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{
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],
"type": "inline_equation",
"content": "\\sigma > \\mathfrak{r}_* / 2"
},
{
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],
"type": "text",
"content": ", then"
}
]
}
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"index": 18
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"type": "interline_equation",
"angle": 0,
"lines": [
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"type": "interline_equation",
"content": "2 \\varepsilon \\geq 1 - P \\left(\\mathcal {M} _ {*} ^ {t _ {*}}\\right) \\geq 1 - P \\left([ - K, K ] ^ {D}\\right) \\geq c _ {2} (K, D, r *)",
"image_path": "4cbd5bf11eb05763a5aefb0710df1ccaf0565a2f07452a1dddb2777c4edc58e1.jpg"
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]
}
],
"index": 19
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{
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"type": "text",
"angle": 0,
"lines": [
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"type": "text",
"content": "where "
},
{
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],
"type": "inline_equation",
"content": "c_{2}"
},
{
"bbox": [
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643,
542,
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],
"type": "text",
"content": " is some positive constant. It is a contradiction following from the smallness of "
},
{
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542,
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],
"type": "inline_equation",
"content": "\\varepsilon"
},
{
"bbox": [
52,
643,
542,
667
],
"type": "text",
"content": ". Let's make a claim that if "
},
{
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52,
643,
542,
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],
"type": "inline_equation",
"content": "\\sigma \\in [2t_{*}, r_{*} / 2]"
},
{
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52,
643,
542,
667
],
"type": "text",
"content": ", then for every "
},
{
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],
"type": "inline_equation",
"content": "y \\in \\mathbb{R}^{D}"
},
{
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],
"type": "text",
"content": ", there is some "
},
{
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],
"type": "inline_equation",
"content": "z \\in \\mathbb{R}^{D}"
},
{
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],
"type": "text",
"content": " such that "
},
{
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],
"type": "inline_equation",
"content": "|z - y|_{2} \\leq \\sigma"
},
{
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],
"type": "text",
"content": " and "
},
{
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],
"type": "inline_equation",
"content": "\\mathcal{B}_{\\sigma / 2}(z) \\cap \\mathcal{M}_{*}^{t_{*}} = \\emptyset"
},
{
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],
"type": "text",
"content": "."
}
]
}
],
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},
{
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],
"type": "text",
"angle": 0,
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"spans": [
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"type": "text",
"content": "Following from the claim, we have"
}
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"index": 21
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"lines": [
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"content": "\\nu \\left(Y + \\epsilon \\notin \\mathcal {M} _ {*} ^ {t _ {*}} | Y = y\\right) \\geq \\nu \\left(\\epsilon \\in \\mathcal {B} _ {\\sigma / 2} (z - y)\\right).",
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"content": "\\widetilde {\\sigma} _ {*} = n ^ {- \\alpha} + n ^ {- \\beta_ {*} / 2 (\\beta_ {*} + t _ {*})} \\asymp \\left\\{ \\begin{array}{l l} n ^ {- \\alpha}, & \\quad \\alpha < \\beta_ {*} / \\{2 (\\beta_ {*} + t _ {*}) \\} \\\\ n ^ {\\beta_ {*} / 2 (\\beta_ {*} + t _ {*})}, & \\quad \\text {o t h e r w i s e}. \\end{array} \\right.",
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"content": "\\left\\{\\left(\\prod_ {j = 1} ^ {r} (1 / M _ {1} - | u ^ {(j)} - u _ {\\boldsymbol {\\ell} ^ {(1)}} ^ {(j)} |) _ {+}\\right) \\left(\\prod_ {j = 1} ^ {r _ {\\prime}} (1 / M _ {2} - | v ^ {(j)} - v _ {\\boldsymbol {\\ell} ^ {(2)}} ^ {(j)} |) _ {+}\\right) \\right\\} _ {\\mathbf {u} _ {\\boldsymbol {\\ell} ^ {(1)}}, \\mathbf {v} _ {\\boldsymbol {\\ell} ^ {(2)}} \\in \\mathbf {D} (M)}.",
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"content": " and "
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"type": "text",
"content": ", and from the use of Lemma 8 there exist "
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"content": "\\mathrm{Mult}_m^{r + r'}"
},
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"type": "text",
"content": " networks in the class"
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"content": "\\mathcal {F} \\left(2 + (m + 5) \\lceil \\log_ {2} (r + r _ {\\prime}) \\rceil , (r + r _ {\\prime}, 2 (r + r _ {\\prime}), r + r _ {\\prime}, 6 (r + r _ {\\prime}), 6 (r + r _ {\\prime}), \\dots , 6 (r + r _ {\\prime}), 1)\\right)",
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"type": "text",
"content": "computing "
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"type": "inline_equation",
"content": "(\\prod_{j=1}^{r}(1 / M_1 - |u^{(j)} - u_{\\ell^{(1)}}|)_+)\\times (\\prod_{j=1}^{r'}(1 / M_2 - |v^{(j)} - v_{\\ell^{(2)}}|)_+)"
},
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"type": "text",
"content": " up to an error that is bounded by "
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"type": "inline_equation",
"content": "(r + r')^2 2^{-m}"
},
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],
"type": "text",
"content": ". Observe that we have two extra hidden layers to compute "
},
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"type": "inline_equation",
"content": "(1 / M_1 - |u^{(j)} - u_{\\ell^{(1)}}|)_+)"
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"type": "text",
"content": " and "
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"type": "inline_equation",
"content": "(1 / M_2 - |v^{(j)} - v_{\\ell^{(2)}}|)_+)"
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"type": "text",
"content": " for fixed "
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"type": "inline_equation",
"content": "\\mathbf{u}_{\\ell^{(1)}}"
},
{
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],
"type": "text",
"content": " and "
},
{
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52,
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],
"type": "inline_equation",
"content": "\\mathbf{v}_{\\ell^{(2)}}"
},
{
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],
"type": "text",
"content": " respectively, before we enter into the multinomial computation by regime invoking Lemma 8. Observe that the number of parameters in this network is upper bounded by "
},
{
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],
"type": "inline_equation",
"content": "37(r + r')^2 (m + 5)\\lceil \\log_2(r + r') \\rceil"
},
{
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],
"type": "text",
"content": "."
}
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}
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{
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"type": "text",
"angle": 0,
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"spans": [
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"type": "text",
"content": "Now we use the parallelization technique to have "
},
{
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"type": "inline_equation",
"content": "(M_1 + 1)^r \\cdot (M_1 + 1)^r"
},
{
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"type": "text",
"content": " parallel architecture for all elements of "
},
{
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"type": "inline_equation",
"content": "\\mathbf{D}(M)"
},
{
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"type": "text",
"content": ". This provides the existence of the network with the number of non-zero parameters bounded by "
},
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],
"type": "inline_equation",
"content": "37(r + r_r)^2 (M_1 + 1)^r (M_2 + 1)^{r'}(m + 5)\\lceil \\log_2(r + r_r)\\rceil"
}
]
}
],
"index": 7
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"type": "text",
"angle": 0,
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"type": "text",
"content": "By Lemma 8, for any "
},
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],
"type": "inline_equation",
"content": "\\mathbf{x} \\in \\mathbb{R}^r"
},
{
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],
"type": "text",
"content": ", "
},
{
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],
"type": "inline_equation",
"content": "\\mathrm{Mult}_m^r(\\mathbf{x}) = 0"
},
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],
"type": "text",
"content": " if one of the components of "
},
{
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],
"type": "inline_equation",
"content": "\\mathbf{x}"
},
{
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],
"type": "text",
"content": " is zero. This shows that for any "
},
{
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],
"type": "inline_equation",
"content": "\\mathbf{u}_{\\ell^{(1)}}, \\mathbf{v}_{\\ell^{(2)}} \\in \\mathbf{D}(M)"
},
{
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],
"type": "text",
"content": ", the support of the function "
},
{
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"type": "inline_equation",
"content": "(\\mathbf{u}, \\mathbf{v}) \\mapsto (\\mathrm{Hat}^{r + r'}(\\mathbf{u}, \\mathbf{v}))_{\\mathbf{u}_{\\ell^{(1)}}, \\mathbf{v}_{\\ell^{(2)}}}"
},
{
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"type": "text",
"content": " is contained in the support of the function "
},
{
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52,
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],
"type": "inline_equation",
"content": "(\\mathbf{u}, \\mathbf{v}) \\mapsto \\left(\\prod_{j=1}^{r} (1 / M - |u^{(j)} - u_{\\ell^{(1)}}^{(j)})|) + \\prod_{j=1}^{r'} (1 / M - |v^{(j)} - v_{\\ell^{(2)}}^{(j)})|\\right)"
},
{
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],
"type": "text",
"content": "."
}
]
}
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},
{
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],
"type": "text",
"angle": 0,
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"spans": [
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"type": "text",
"content": "Proof of Theorem 5. All the constructed networks in this proof are of the form "
},
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"type": "inline_equation",
"content": "\\mathcal{F}(L,\\mathbf{p},s) = \\mathcal{F}(L,\\mathbf{p},s,\\infty)"
},
{
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],
"type": "text",
"content": " with "
},
{
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],
"type": "inline_equation",
"content": "F = \\infty"
},
{
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],
"type": "text",
"content": ". Denote "
},
{
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],
"type": "inline_equation",
"content": "M_{1} = M^{\\widetilde{\\beta} /\\beta}"
},
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],
"type": "text",
"content": ", "
},
{
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],
"type": "inline_equation",
"content": "M_2 = M^{\\widetilde{\\beta} /\\beta '}"
},
{
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],
"type": "text",
"content": ", "
},
{
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"type": "inline_equation",
"content": "\\beta_{\\mathrm{sum}} = \\beta +\\beta_{t}"
},
{
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],
"type": "text",
"content": ", and "
},
{
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],
"type": "inline_equation",
"content": "\\mathsf{r}_{\\mathrm{sum}} = r + r_{t}"
},
{
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],
"type": "text",
"content": ". Let "
},
{
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],
"type": "inline_equation",
"content": "M"
},
{
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],
"type": "text",
"content": " be the largest integer such that "
},
{
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"type": "inline_equation",
"content": "\\mathsf{M} = (M_1 + 1)^r (M_2 + 1)^{r'}\\leq N"
},
{
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],
"type": "text",
"content": " and define "
},
{
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],
"type": "inline_equation",
"content": "L^{*}\\coloneqq (m + 5)\\lceil \\log_{2}(\\beta_{\\mathrm{sum}}\\lor \\mathsf{r}_{\\mathrm{sum}})\\rceil"
},
{
"bbox": [
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],
"type": "text",
"content": ". Thanks to (34), (33) and Lemma 9, we can add one hidden layer to the network "
},
{
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],
"type": "inline_equation",
"content": "\\mathrm{Mon}_{m,\\beta_{\\mathrm{sum}}}^{\\mathsf{r}_{\\mathrm{sum}}}"
},
{
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"type": "text",
"content": " to obtain a network"
}
]
}
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"angle": 0,
"lines": [
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"spans": [
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"type": "interline_equation",
"content": "Q _ {1} \\in \\mathcal {F} \\left(2 + L ^ {*}, (r, 6 \\lceil \\beta \\rceil C _ {\\mathrm {r} _ {\\text {s u m}}, \\beta_ {\\text {s u m}}}, \\dots , 6 \\lceil \\beta \\rceil C _ {\\mathrm {r} _ {\\text {s u m}}, \\beta_ {\\text {s u m}}}, C _ {\\mathrm {r} _ {\\text {s u m}}, \\beta_ {\\text {s u m}}}, \\mathrm {M})\\right),",
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"type": "text",
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"type": "text",
"content": "such that "
},
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"type": "inline_equation",
"content": "Q_{1}(\\mathbf{u},\\mathbf{v})\\in [0,1]^{\\mathsf{M}}"
},
{
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"type": "text",
"content": " and for any "
},
{
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"type": "inline_equation",
"content": "\\mathbf{u}\\in [0,1]^{r}"
},
{
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"type": "text",
"content": " and for any "
},
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"type": "inline_equation",
"content": "\\mathbf{v}\\in [0,1]^{r'}"
}
]
}
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"index": 11
},
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"type": "interline_equation",
"angle": 0,
"lines": [
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"spans": [
{
"bbox": [
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"type": "interline_equation",
"content": "\\left| Q _ {1} (\\mathbf {u}, \\mathbf {v}) - \\left(\\frac {P ^ {\\beta , \\beta_ {\\prime}} f (\\mathbf {u} , \\mathbf {v})}{B} + \\frac {1}{2}\\right) _ {\\mathbf {u} _ {\\ell^ {(1)}}, \\mathbf {v} _ {\\ell^ {(2)}} \\in \\mathbf {D} (M)} \\right| _ {\\infty} \\leq \\beta_ {\\text {s u m}} ^ {2} 2 ^ {- m} \\tag {36}",
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"type": "text",
"angle": 0,
"lines": [
{
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"spans": [
{
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],
"type": "text",
"content": "with "
},
{
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],
"type": "inline_equation",
"content": "B \\coloneqq \\lceil 2Ke^{\\mathbf{r}_{\\mathrm{sum}}}\\rceil"
},
{
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],
"type": "text",
"content": ". The total number of non-zero parameters in the "
},
{
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],
"type": "inline_equation",
"content": "Q_{1}"
},
{
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],
"type": "text",
"content": " network is "
},
{
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],
"type": "inline_equation",
"content": "6\\mathrm{r}_{\\mathrm{sum}}(\\beta_{\\mathrm{sum}} + 1)C_{\\mathrm{r}_{\\mathrm{sum}},\\beta_{\\mathrm{sum}}} + 42(\\beta_{\\mathrm{sum}} + 1)^{2}C_{\\mathrm{r}_{\\mathrm{sum}},\\beta_{\\mathrm{sum}}}^{2}(L^{*} + 1) + C_{\\mathrm{r}_{\\mathrm{sum}},\\beta_{\\mathrm{sum}}}\\mathsf{M}"
},
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],
"type": "text",
"content": "."
}
]
}
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"type": "text",
"angle": 0,
"lines": [
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"spans": [
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"type": "text",
"content": "Recall that the network "
},
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],
"type": "inline_equation",
"content": "\\mathrm{Hat}^{\\mathbf{r}_{\\mathrm{sum}}}"
},
{
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],
"type": "text",
"content": " computes the products of hat functions (splines) "
},
{
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],
"type": "inline_equation",
"content": "(\\prod_{j=1}^{r}(1/M_1 - |u^{(j)} - u_{\\ell^{(1)}}|)_+)(\\prod_{j=1}^{r'}(1/M_2 - |v^{(j)} - v_{\\ell^{(2)}}|)_+)"
},
{
"bbox": [
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543,
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],
"type": "text",
"content": " up to an error that is bounded by "
},
{
"bbox": [
52,
586,
543,
661
],
"type": "inline_equation",
"content": "\\mathbf{r}_{\\mathrm{sum}}^2 2^{-m}"
},
{
"bbox": [
52,
586,
543,
661
],
"type": "text",
"content": ". It requires at most "
},
{
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],
"type": "inline_equation",
"content": "37\\mathbf{r}_{\\mathrm{sum}}^2 NL^*"
},
{
"bbox": [
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543,
661
],
"type": "text",
"content": " active parameters. Observe that "
},
{
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],
"type": "inline_equation",
"content": "C_{\\mathbf{r}_{\\mathrm{sum}}, \\beta_{\\mathrm{sum}}} \\leq (\\beta_{\\mathrm{sum}} + 1)^{\\mathbf{r}_{\\mathrm{sum}}} \\leq N"
},
{
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],
"type": "text",
"content": " by the definition of "
},
{
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],
"type": "inline_equation",
"content": "C_{r,\\beta}"
},
{
"bbox": [
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],
"type": "text",
"content": " and the assumptions on "
},
{
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],
"type": "inline_equation",
"content": "N"
},
{
"bbox": [
52,
586,
543,
661
],
"type": "text",
"content": ". By Lemma 6, the networks "
},
{
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543,
661
],
"type": "inline_equation",
"content": "Q_1"
},
{
"bbox": [
52,
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661
],
"type": "text",
"content": " and "
},
{
"bbox": [
52,
586,
543,
661
],
"type": "inline_equation",
"content": "\\mathrm{Hat}^{\\mathbf{r}_{\\mathrm{sum}}}"
},
{
"bbox": [
52,
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661
],
"type": "text",
"content": " can be embedded into a joint parallel network "
},
{
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],
"type": "inline_equation",
"content": "(Q_1, \\mathrm{Hat}^{\\mathbf{r}_{\\mathrm{sum}}})"
},
{
"bbox": [
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],
"type": "text",
"content": " with "
},
{
"bbox": [
52,
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],
"type": "inline_equation",
"content": "2 + L^*"
},
{
"bbox": [
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],
"type": "text",
"content": " hidden layers of size "
},
{
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],
"type": "inline_equation",
"content": "(\\mathbf{r}_{\\mathrm{sum}}, 6(\\mathbf{r}_{\\mathrm{sum}} + \\lceil \\beta_{\\mathrm{sum}} \\rceil)N, \\ldots, 6(\\mathbf{r}_{\\mathrm{sum}} + \\lceil \\beta_{\\mathrm{sum}} \\rceil)N, 2M)"
},
{
"bbox": [
52,
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661
],
"type": "text",
"content": ". Using "
},
{
"bbox": [
52,
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661
],
"type": "inline_equation",
"content": "C_{r,\\beta} \\vee (M + 1)^r \\leq N"
},
{
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"content": "Lemma 7. For any positive integer "
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