changing draft-pdf.yml

paper.md

@@ -27,16 +27,16 @@ Implicit generative models have the capability to learn arbitrary complex data d
 
 In addition to its core functionalities, `ISL` offers a suite of utility functions, including support for generative adversarial networks and tools for time series analysis and generation. This makes `ISL` a valuable resource for both machine learning researchers and data scientists/software developers who seek to train their models with this novel approach.
 
-`ISL` has already contributed in a scientific publications [de2024training], underscoring its utility and impact in the field. Its combination of speed, thoughtful design, and robust machine learning functionalities for Implicit Generative Models positions `ISL` as a powerful tool for advancing scientific research and practical applications alike in the area of Implicit Generative Models.
+`ISL` has already contributed in a scientific publications [@de2024training], underscoring its utility and impact in the field. Its combination of speed, thoughtful design, and robust machine learning functionalities for Implicit Generative Models positions `ISL` as a powerful tool for advancing scientific research and practical applications alike in the area of Implicit Generative Models.
 
 # Methods
 
 Implicit generative models employ an $m$-dimensional latent random variable (r.v.) $\mathbf{z}$ to simulate random samples from a prescribed $n$-dimensional target probability distribution. To be precise, the latent variable undergoes a transformation through a deterministic function $g_{\theta}$, which maps $\mathbb{R}^m \mapsto \mathbb{R}^n$ using the parameter set $\theta$. Given the model capability to generate samples with ease, various techniques can be employed for contrasting two sample collections: one originating from the genuine data distribution and the other from the model distribution. This approach essentially constitutes a methodology for the approximation of probability distributions via comparison.
-Generative adversarial networks (GANs) [goodfellow2014generative], $f$-GANs [nowozin2016f], Wasserstein-GANs (WGANs) [arjovsky2017wasserstein], adversarial variational Bayes (AVB) [mescheder2017adversarial], and maximum mean-miscrepancy (MMD) GANs [li2017mmd] are some popular methods that fall within this framework. 
+Generative adversarial networks (GANs) [@goodfellow2014generative], $f$-GANs [@nowozin2016f], Wasserstein-GANs (WGANs) [@arjovsky2017wasserstein], adversarial variational Bayes (AVB) [@mescheder2017adversarial], and maximum mean-miscrepancy (MMD) GANs [li2017mmd] are some popular methods that fall within this framework. 
 
-Approximation of 1-dimensional (1D) parametric distributions is a seemingly naive problem for which the above-mentioned models can perform below expectations. In [zaheer2017gan], the authors report that various types of GANs struggle to approximate relatively simple distributions from samples, emerging with MMD-GAN as the most promising technique. However, the latter implements a kernelized extension of a moment-matching criterion defined over a reproducing kernel Hilbert space, and consequently, the objective function is expensive to compute. 
+Approximation of 1-dimensional (1D) parametric distributions is a seemingly naive problem for which the above-mentioned models can perform below expectations. In [@zaheer2017gan], the authors report that various types of GANs struggle to approximate relatively simple distributions from samples, emerging with MMD-GAN as the most promising technique. However, the latter implements a kernelized extension of a moment-matching criterion defined over a reproducing kernel Hilbert space, and consequently, the objective function is expensive to compute. 
 
-In this work, we introduce a novel approach to train univariate implicit models that relies on a fundamental property of rank statistics. Let $r_1 < r_2 < \cdots < r_k$ be a ranked (ordered) sequence of independent and identically distributed (i.i.d.) samples from the generative model with probability density function (pdf) $\tilde{p}$, and let $y$ be a random sample from a pdf $p$. If $\tilde{p} = p$, then $\mathbb{P}(r_{i-1} \leq y < r_{i}) = \frac{1}{K}$ for every $i = 1, \ldots, K+1$, with the convention that $r_0=-\infty$ and $r_{K+1}=\infty$ (see, e.g., [rosenblatt1952remarks] or [elvira2016adapting] for a short explicit proof). This invariant property holds for any continuously distributed data, i.e., for any data with a pdf $p$. Consequently, even if $p$ is unknown, we can leverage this invariance to construct an objective (loss) function.  This objective function eliminates the need for a discriminator, directly measuring the discrepancy of the transformed samples with respect to (w.r.t.) the uniform distribution. The computational cost of evaluating this loss increases linearly with both $K$ and $N$, allowing for low-complexity mini-batch updates. Moreover, the proposed criterion is invariant across true data distributions, hence we refer to the resulting objective function as invariant statistical loss (ISL). Because of this property, the ISL can be exploited to learn multiple modes in mixture models and different time steps when learning temporal processes. Additionally, considering the marginal distributions independently, it is straightforward to extend ISL to the multivariate case.
+In this work, we introduce a novel approach to train univariate implicit models that relies on a fundamental property of rank statistics. Let $r_1 < r_2 < \cdots < r_k$ be a ranked (ordered) sequence of independent and identically distributed (i.i.d.) samples from the generative model with probability density function (pdf) $\tilde{p}$, and let $y$ be a random sample from a pdf $p$. If $\tilde{p} = p$, then $\mathbb{P}(r_{i-1} \leq y < r_{i}) = \frac{1}{K}$ for every $i = 1, \ldots, K+1$, with the convention that $r_0=-\infty$ and $r_{K+1}=\infty$ (see, e.g., [@rosenblatt1952remarks] or [@elvira2016adapting] for a short explicit proof). This invariant property holds for any continuously distributed data, i.e., for any data with a pdf $p$. Consequently, even if $p$ is unknown, we can leverage this invariance to construct an objective (loss) function.  This objective function eliminates the need for a discriminator, directly measuring the discrepancy of the transformed samples with respect to (w.r.t.) the uniform distribution. The computational cost of evaluating this loss increases linearly with both $K$ and $N$, allowing for low-complexity mini-batch updates. Moreover, the proposed criterion is invariant across true data distributions, hence we refer to the resulting objective function as invariant statistical loss (ISL). Because of this property, the ISL can be exploited to learn multiple modes in mixture models and different time steps when learning temporal processes. Additionally, considering the marginal distributions independently, it is straightforward to extend ISL to the multivariate case.
 
 
 # Software Description