In this task, you are given the abstract of a research paper. Your task is to generate a summary of this abstract. Your summary should not be very short, but it's better if it's not more than 30 words.

Understanding the behavior of  stochastic gradient descent (SGD) in the context of deep neural networks has raised lots of concerns recently. Along this line, we theoretically study a general form of gradient based optimization dynamics with unbiased noise, which unifies SGD and standard Langevin dynamics. Through investigating this general optimization dynamics, we analyze the behavior of SGD on escaping from minima and its regularization effects. A novel indicator is derived to characterize the efficiency of escaping from minima through measuring the alignment of noise covariance and the curvature of loss function. Based on this indicator, two conditions are established to show which type of noise structure is superior to isotropic noise in term of escaping efficiency. We further show that the anisotropic noise in SGD satisfies the two conditions, and thus helps to  escape from sharp and poor minima effectively, towards more stable and flat minima that typically generalize well. We verify our understanding through comparing
 this anisotropic diffusion with full gradient descent plus isotropic diffusion (i.e. Langevin dynamics) and other types of position-dependent noise.
We provide theoretical and empirical analysis on the role of anisotropic noise introduced by stochastic gradient on escaping from minima.

Clustering is a fundamental machine learning method. The quality of its results is dependent on the data distribution. For this reason, deep neural networks can be used for learning better representations of the data. In this paper, we propose a systematic taxonomy for clustering with deep learning, in addition to a review of methods from the field. Based on our taxonomy, creating new methods is more straightforward. We also propose a new approach which is built on the taxonomy and surpasses some of the limitations of some previous work. Our experimental evaluation on image datasets shows that the method approaches state-of-the-art clustering quality, and performs better in some cases.
Unifying framework to perform clustering using deep neural networks

Extending models with auxiliary latent variables is a well-known technique to in-crease model expressivity. Bachman & Precup (2015); Naesseth et al. (2018); Cremer et al. (2017); Domke & Sheldon (2018) show that Importance Weighted Autoencoders (IWAE) (Burda et al., 2015) can be viewed as extending the variational family with auxiliary latent variables. Similarly, we show that this view encompasses many of the recent developments in variational bounds (Maddisonet al., 2017; Naesseth et al., 2018; Le et al., 2017; Yin & Zhou, 2018; Molchanovet al., 2018; Sobolev & Vetrov, 2018). The success of enriching the variational family with auxiliary latent variables motivates applying the same techniques to the generative model. We develop a generative model analogous to the IWAE bound and empirically show that it outperforms the recently proposed Learned Accept/Reject Sampling algorithm (Bauer & Mnih, 2018), while being substantially easier to implement. Furthermore, we show that this generative process provides new insights on ranking Noise Contrastive Estimation (Jozefowicz et al.,2016; Ma & Collins, 2018) and Contrastive Predictive Coding (Oord et al., 2018).

Monte Carlo Objectives are analyzed using auxiliary variable variational inference, yielding a new analysis of CPC and NCE as well as a new generative model.