Expected Information Maximization: Using the I-Projection for Mixture Density Estimation
Modelling highly multi-modal data is a challenging problem in machine learning. Most algorithms are based on maximizing the likelihood, which corresponds to the M(oment)-projection of the data distribution to the model distribution. The M-Projection forces the model to average over modes that can not be represented by the model. In contrast, the I(information)-Projection ignores such modes in the data and concentrates on the modes the model can represent. Such behavior is appealing whenever we deal with highly multi-modal data where it is more important to model single modes correctly instead of covering all the modes. Despite this advantage, the I-projection is rarely used in practise due to the lack of algorithms that can efficiently optimize it based on data. In this work, we present a new algorithm called Expected Information Maximization (EIM) for computing the I-projection solely based on samples for general latent variable models, where we focus on Gaussian mixtures models and Gaussian mixture of experts. Our approach applies a variational upper bound to the I-projection objective which decomposes the original objective into single objectives for each mixture component as well as for the coefficients, allowing an efficient optimization. Similar to GANs, our approach also employs discriminators but uses a more stable optimization procedures optimizing a tight upper bound. We show that our algorithm is much more effective in computing the I-projection than recent GAN approaches and we illustrate the effectiveness of our approach for modelling multi-modal behavior on two pedestrian and traffic prediction datasets.
This paper propose EIM an analog to EM but to perform the I-projection (i.e. reverse-KL) instead of the usual M-projection for EM. The motivation is that the reverse-KL is mode-seeking in contrast to the forward-KL which is mode-covering. The authors argue that in the case that the model is mis-specified, I-projection is sometimes desired as to avoid putting mass on very unlikely regions of the space under the target p. The authors propose an iterative procedure that alternates between estimating likelihood ratios and proposal distribution by minimizing an upper bound on the reverse-KL. The derivations seem correct. There are some experiments, majoritarily in the robotics domain. As the author point out, likelihood shouldn*t be the right metric since you are now minimizing the reverse-KL---I would have liked the authors to spend some more time on the right way to evaluate---and actually use that new metric. Finally, there has been plethora of work on different objectives and distance between distributions as well as a zoo of lower/upper bounds on how to evaluate them---it would be interesting to have more connections to prior work. [Pros] - clearly written - clear motivation - correct derivations - interesting algorithm [Cons] - experiments are a little weak (and focus on a single domain) - would have liked to see an explicit algorithm for the optimization procedure - small lack of clarity in the presentation of Section 4.1---notation q_t is not introduced for example - more discussion about the evaluation metric - linking it more to prior work
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Expected Information Maximization: Using the I-Projection for Mixture Density Estimation
Modelling highly multi-modal data is a challenging problem in machine learning. Most algorithms are based on maximizing the likelihood, which corresponds to the M(oment)-projection of the data distribution to the model distribution. The M-Projection forces the model to average over modes that can not be represented by the model. In contrast, the I(information)-Projection ignores such modes in the data and concentrates on the modes the model can represent. Such behavior is appealing whenever we deal with highly multi-modal data where it is more important to model single modes correctly instead of covering all the modes. Despite this advantage, the I-projection is rarely used in practise due to the lack of algorithms that can efficiently optimize it based on data. In this work, we present a new algorithm called Expected Information Maximization (EIM) for computing the I-projection solely based on samples for general latent variable models, where we focus on Gaussian mixtures models and Gaussian mixture of experts. Our approach applies a variational upper bound to the I-projection objective which decomposes the original objective into single objectives for each mixture component as well as for the coefficients, allowing an efficient optimization. Similar to GANs, our approach also employs discriminators but uses a more stable optimization procedures optimizing a tight upper bound. We show that our algorithm is much more effective in computing the I-projection than recent GAN approaches and we illustrate the effectiveness of our approach for modelling multi-modal behavior on two pedestrian and traffic prediction datasets.
user
<think>Okay, to do a decent peer review, I should start with summarizing what the work is about. This includes novelty, contributions, clarity, and impact. I might include suggestions to issues that I find critical to the usefulness of the work. Starting with novelty, there are different types of novelty. The first aspect of novelty is scope and scale. Is the work covering a broader perspective than previous works? The second aspect is methodology. Is the proposed methodology novel compared to previous methods? Do they provide any significant advantage over previous methodologies? The third aspect is results. Are the results different compared previous work? Does this lead to any novel theories or an extension of current theories? Next is clarity, I will be reading the work in depth and analyze if any sentences can be potentially ambiguous or misleading. If any sentences are ambiguous, I will state them and give a suggestion on how to rephrase such sentence or how to make more clear to the audience. Lastly, impact is important. I will attempt to predict how such work will impact the scientific community. This might include the applications of such work and how the work might transform the way we study a certain topic. Finally, I will might only mention issues that are critical. This is to keep the review as concise and helpful as possible.</think>



This paper propose EIM an analog to EM but to perform the I-projection (i.e. reverse-KL) instead of the usual M-projection for EM. The motivation is that the reverse-KL is mode-seeking in contrast to the forward-KL which is mode-covering. The authors argue that in the case that the model is mis-specified, I-projection is sometimes desired as to avoid putting mass on very unlikely regions of the space under the target p. The authors propose an iterative procedure that alternates between estimating likelihood ratios and proposal distribution by minimizing an upper bound on the reverse-KL. The derivations seem correct. There are some experiments, majoritarily in the robotics domain. As the author point out, likelihood shouldn*t be the right metric since you are now minimizing the reverse-KL---I would have liked the authors to spend some more time on the right way to evaluate---and actually use that new metric. Finally, there has been plethora of work on different objectives and distance between distributions as well as a zoo of lower/upper bounds on how to evaluate them---it would be interesting to have more connections to prior work. [Pros] - clearly written - clear motivation - correct derivations - interesting algorithm [Cons] - experiments are a little weak (and focus on a single domain) - would have liked to see an explicit algorithm for the optimization procedure - small lack of clarity in the presentation of Section 4.1---notation q_t is not introduced for example - more discussion about the evaluation metric - linking it more to prior work
assistant
This paper propose EIM an analog to EM but to perform the I-projection (i.e. reverse-KL) instead of the usual M-projection for EM. The motivation is that the reverse-KL is mode-seeking in contrast to the forward-KL which is mode-covering. The authors argue that in the case that the model is mis-specified, I-projection is sometimes desired as to avoid putting mass on very unlikely regions of the space under the target p. The authors propose an iterative procedure that alternates between estimating likelihood ratios and proposal distribution by minimizing an upper bound on the reverse-KL. The derivations seem correct. There are some experiments, majoritarily in the robotics domain. As the author point out, likelihood shouldn*t be the right metric since you are now minimizing the reverse-KL---I would have liked the authors to spend some more time on the right way to evaluate---and actually use that new metric. Finally, there has been plethora of work on different objectives and distance between distributions as well as a zoo of lower/upper bounds on how to evaluate them---it would be interesting to have more connections to prior work. [Pros] - clearly written - clear motivation - correct derivations - interesting algorithm [Cons] - experiments are a little weak (and focus on a single domain) - would have liked to see an explicit algorithm for the optimization procedure - small lack of clarity in the presentation of Section 4.1---notation q_t is not introduced for example - more discussion about the evaluation metric - linking it more to prior work