Robustness of Invertible Neural Networks

在低维中构建直方图或概率分布非常简单，作为人类，我们能够在头脑中以一维，二维甚至三维的方式可视化它们。然而，真实的世界往往不局限于低维度，事件发生的概率可能取决于大量因素，从而导致高维概率分布。然而，由于问题的高维性，通常不可能以标准方式构建这样的直方图。可逆神经网络（INN）和规范化流（NF）提供了一种评估复杂高维分布可能性的方法。这是通过学习一个简单的潜在参数分布（简单，我们知道）和真实分布（我们不知道，很难建模）之间的非线性双射映射来实现的。然后，数据样本的似然性等于由归一化常数缩放的简单潜在分布上的映射数据点的似然性。因此，我们不是测量真实分布的可能性，而是将其映射到一个潜在分布，在那里我们可以轻松地评估它，然后适当地缩放它。此外，INN允许我们从潜在空间中采样并执行逆映射以生成数据样本。虽然支撑这些方法的理论是有吸引力的，但关于其可用性和准确性仍然存在一些开放的问题。解决这些问题将是我的论文的核心，也是我研究的主要内容。第一个主要问题是计算评估数据精确似然性所需的归一化常数。简单的计算在计算上是非常昂贵的，因此需要利用加速计算的方法。标准方法对网络施加了各种约束，可能会损害映射的表现力。解决这个问题是我的论文的第一个主要目标，已经取得了重大进展，解决上述问题的方法已经实施，并等待出版。第二个问题是强制映射的可逆性（以生成样本），这个问题构成了为给定输出推断输入的更广泛的画面的一部分。对于这种应用，可以采用各种优化方法来获得输入，但是它们通常不起作用，有时不是全部。这是目前形成的大部分时间和初步的方法已经过测试，但没有一个成功的工作在所有可能的映射。第三，我将对INN和NF的稳定性和鲁棒性进行彻底的调查。最近的研究表明，这些方法可能并不像它们看起来那么准确。因此，我正在进行的工作的一部分是调查，这些方法在哪里以及如何失败。因此，我们有可能弥补他们的缺陷。以上三点是目前我的研究和项目工作的主要内容。从文献来看，它们都有明确的联系和良好的动机。解决所有这些问题将为社区作出重大贡献。

Constructing histograms or probability distributions in low dimensions is very simple, and as humans we are able to visualise them in our heads in one, two or maybe even three dimensions. However, the real world is often not constrained to low dimensions, and the probability of an event occurring can be dependent on a large number of factors, resulting in a high-dimensional probability distribution. However, constructing such a histogram in a standard way is often not possible due to the high dimensionality of the problem. Invertible Neural Networks (INN) and Normalising Flows (NFs) offer a way to evaluate the likelihoods of complex high-dimensional distributions. This is achieved by learning a non-linear bijective mapping between a simple latent parametric distribution (which is simple, and we know) and the true distribution (which we don't know and is hard to model). The likelihood of a data sample is then equal to the likelihood of the mapped data point on the simple latent distribution scaled by a normalising constant. Thus, rather than measuring the likelihood on the true distribution, we instead map it to a latent distribution where we can evaluate it easily and then scale it appropriately. Furthermore, INNs allow us to sample from the latent space and perform the inverse mapping to generate data samples. Whilst the theory underpinning these methods is attractive, several open issues remain regarding their usability and accuracy. Addressing these issues will form the core of my thesis and the majority or my research. The first major problem is calculating the normalising constant needed to evaluate the exact likelihood of the data. Naïve calculation is computationally prohibitively expensive, and as such methods to speed up the computation need to leveraged. Standard methods enforce a variety of constraints on the network, potentially harming the expressivity of the mapping. Addressing this issue is the first major objective for my thesis, significant progress has already been made and a method addressing the above issues has been implemented and is awaiting publication. The second issue is enforcing the invertibility of the mapping (to generate samples), this issue forms part of a much broader picture of inferring in input for a given output. For this application, various optimization methods can be employed to obtain the input, however they often do not work well and sometimes not all. This is currently forming the majority of time and preliminary methods have been tested, but none that work successfully across all possible mappings. Thirdly, I will be constructing a thorough investigation into the stability and robustness of INNs and NFs. Recent work suggests that these methods may not be as accurate as they seem. As such part of my ongoing work is to investigate, where and how these methods fail. Thus, allowing us to potentially compensate for their pitfalls. The above three points are currently forming the majority of my research and project work. They are all clearly linked and well-motivated from the literature. Addressing all of them will provide a significant contribution to the community.