Locally Linear Embedding (LLE) is a nonlin- ear spectral dimensionality reduction and manifold learning method. It has two main steps which are linear reconstruc- tion and linear embedding of points in the input space and embedding space, respectively. In this work, we propose two novel generative versions of LLE, named Generative LLE (GLLE), whose linear reconstruction steps are stochastic rather than deterministic. GLLE assumes that every data point is caused by its linear reconstruction weights as latent factors. The proposed GLLE algorithms can generate various LLE embeddings stochastically while all the generated embeddings relate to the original LLE embedding. We propose two versions for stochastic linear reconstruction, one using expectation maximization and another with direct sampling from a derived distribution by optimization. The proposed GLLE methods are closely related to and inspired by variational inference, factor analysis, and probabilistic principal component analysis. Our simulations show that the proposed GLLE methods work effectively in unfolding and generating submanifolds of data.

We analyze the effect of offline and online triplet mining for colorectal cancer (CRC) histopathology dataset containing 100,000 patches. We consider the extreme, i.e., farthest and nearest patches with respect to a given anchor, both in online and offline mining. While many works focus solely on how to select the triplets online (batch-wise), we also study the effect of extreme distances and neighbor patches before training in an offline fashion. We analyze the impacts of extreme cases for offline versus online mining, including easy positive, batch semi-hard, and batch hard triplet mining as well as the neighborhood component analysis loss, its proxy version, and distance weighted sampling. We also investigate online approaches based on extreme distance and comprehensively compare the performance of offline and online mining based on the data patterns and explain offline mining as a tractable generalization of the online mining with large mini-batch size. As well, we discuss the relations of different colorectal tissue types in terms of extreme distances. We found that offline mining can generate a better statistical representation of the population by working on the whole dataset.

The computational challenges in turbulent combustion simulations stem from the physical complexities and multi-scale nature of the problem which make it intractable to compute scale-resolving simulations. For most engineering applications, the large scale separation between the flame (typically sub-millimeter scale) and the characteristic turbulent flow (typically centimeter or meter scale)  allows us to evoke simplifying assumptions--such as done for the flamelet model--to pre-compute all the chemical reactions and map them to a low-order manifold. The resulting manifold is then tabulated and looked-up at run-time. As the physical complexity of combustion simulations increases (including radiation, soot formation, pressure variations etc.) the dimensionality of the resulting manifold grows which impedes an efficient tabulation and look-up. In this paper we present a novel approach to model the multi-dimensional combustion manifold. We approximate the combustion manifold using a neural network function approximator and use it to predict the temperature and composition of the reaction. We present a novel training procedure which is developed to generate a smooth output curve for temperature over the course of a reaction. We then evaluate our work against the current approach of tabulation with linear interpolation in combustion simulations. We also provide an ablation study of our training procedure in the context of over-fitting in our model. The combustion dataset used for the modeling of combustion of H2 and O2 in this work is
released alongside this paper. See the poster version here.