Multi-sample, importance-weighted variational autoencoders (IWAE) give tighter bounds and more accurate uncertainty estimates than variational autoencoders (VAEs) trained with a standard single-sample objective. However, IWAEs scale poorly: as the latent dimensionality grows, they require exponentially many samples to retain the benefits of importance weighting. While sequential Monte-Carlo (SMC) can address this problem, it is prohibitively slow because the resampling step imposes sequential structure which cannot be parallelised, and moreover, resampling is non-differentiable which is problematic when learning approximate posteriors. To address these issues, we developed tensor Monte-Carlo (TMC) which gives exponentially many importance samples by separately drawing $K$ samples for each of the $n$ latent variables, then averaging over all $K^n$ possible combinations. While the sum over exponentially many terms might seem to be intractable, in many cases it can be computed efficiently as a series of tensor inner-products. We show that TMC is superior to IWAE on a generative model with multiple stochastic layers trained on the MNIST handwritten digit database, and we show that TMC can be combined with standard variance reduction techniques.

These methods implicitly improve the approximate posterior by drawing multiple samples from a proposal, and resampling to discard samples that do not fit the data (Cremer et al.,

We thank the reviewers for their insightful comments. Due to the space limit, we try our best to answer all the major questions from each reviewer. Please refer to Q1 of reviewer 2 for more details. TMC works even if the observation is chaotic. We will revise the paper to clarify this point.

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We would like to thank the reviewers for their kind and thoughtful comments. Any attempt to mitigate particle degeneracy (e.g. Replicating Fig 1BD, we find similar, albeit less extreme results, with TMC always being faster than SMC. In particular, we have included Eq. 36 in the main text, and also included the corresponding choice of This should help to clarify that Eq. 11 applies to any directed graphical model (we have also included references In the example in Figure 1, we consider a model that does not have a chain-structure (see Appendix Figure 1A). IW AE performs arbitrarily badly due to the high-dimensionality of the state-space.

At run time, a group of agents interact with each other in a shared environment.

Digital Humanities (DH) is an interdisciplinary field that integrates computational methods with humanities scholarship to investigate innovative topics. Each academic discipline follows a unique developmental path shaped by the topics researchers investigate and the methods they employ. With the help of bibliometric analysis, most of previous studies have examined DH across multiple dimensions such as research hotspots, co-author networks, and institutional rankings. However, these studies have often been limited in their ability to provide deep insights into the current state of technological advancements and topic development in DH. As a result, their conclusions tend to remain superficial or lack interpretability in understanding how methods and topics interrelate in the field. To address this gap, this study introduced a new concept of Topic-Method Composition (TMC), which refers to a hybrid knowledge structure generated by the co-occurrence of specific research topics and the corresponding method. Especially by analyzing the interaction between TMCs, we can see more clearly the intersection and integration of digital technology and humanistic subjects in DH. Moreover, this study developed a TMC-based workflow combining bibliometric analysis, topic modeling, and network analysis to analyze the development characteristics and patterns of research disciplines. By applying this workflow to large-scale bibliometric data, it enables a detailed view of the knowledge structures, providing a tool adaptable to other fields.