Likelihood-based Permutation Invariant Loss Function for Probability Distributions
We propose a permutation-invariant loss function designed for the neural networks reconstructing a set of elements without considering the order within its vector representation. Unlike popular approaches for encoding and decoding a set, our work does not rely on a carefully engineered network topology nor by any additional sequential algorithm. The proposed method, Set Cross Entropy, has a natural information-theoretic interpretation and is related to the metrics defined for sets. We evaluate the proposed approach in two object reconstruction tasks and a rule learning task.
 In the manuscript entitled "Likelihood-based Permutation Invariant Loss Function for Probability Distributions" the authors propose a loss function for training against instances in which ordering within the data vector is unimportant. I do not find the proposed loss function to be well motivated, find a number of confusing points (errors?) in the manuscript, and do not easily follow what was done in the examples. First, it should be noted that this is a very restricted consideration of what it means to compare two sets since only sets of equal size are under consideration; this is fundamentally different to the ambitions of e.g. the Hausdorff measure as used in analysis. The logsumexp formulation of the proposed measure is unsatisfactory to me as it directly averages over each of the independent probabilities that a given element is a member of the target set, rather than integrating over the combinatorial set of probabilities for each set of complete possible matches. Moreover, the loss function H() is not necessarily representative of a generative distribution. The definition of the Hausdorff distance given is directional and is therefore not a metric, contrary to what is stated on page 2. I find the description of the problem domain confusing on page 3: the space [0,1]^NxF is described as binary, but then values of log y_i and log (1-y_i) are computed with y in [0,1] so we must imagine these are in fact elements in the open set of reals: (0,1). Clarity of the examples could be greatly improved, in particular by explaining precisely what is the objective of each task and what are the *ingredients* we begin with.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Likelihood-based Permutation Invariant Loss Function for Probability Distributions
We propose a permutation-invariant loss function designed for the neural networks reconstructing a set of elements without considering the order within its vector representation. Unlike popular approaches for encoding and decoding a set, our work does not rely on a carefully engineered network topology nor by any additional sequential algorithm. The proposed method, Set Cross Entropy, has a natural information-theoretic interpretation and is related to the metrics defined for sets. We evaluate the proposed approach in two object reconstruction tasks and a rule learning task.
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>



 In the manuscript entitled "Likelihood-based Permutation Invariant Loss Function for Probability Distributions" the authors propose a loss function for training against instances in which ordering within the data vector is unimportant. I do not find the proposed loss function to be well motivated, find a number of confusing points (errors?) in the manuscript, and do not easily follow what was done in the examples. First, it should be noted that this is a very restricted consideration of what it means to compare two sets since only sets of equal size are under consideration; this is fundamentally different to the ambitions of e.g. the Hausdorff measure as used in analysis. The logsumexp formulation of the proposed measure is unsatisfactory to me as it directly averages over each of the independent probabilities that a given element is a member of the target set, rather than integrating over the combinatorial set of probabilities for each set of complete possible matches. Moreover, the loss function H() is not necessarily representative of a generative distribution. The definition of the Hausdorff distance given is directional and is therefore not a metric, contrary to what is stated on page 2. I find the description of the problem domain confusing on page 3: the space [0,1]^NxF is described as binary, but then values of log y_i and log (1-y_i) are computed with y in [0,1] so we must imagine these are in fact elements in the open set of reals: (0,1). Clarity of the examples could be greatly improved, in particular by explaining precisely what is the objective of each task and what are the *ingredients* we begin with.
assistant
 In the manuscript entitled "Likelihood-based Permutation Invariant Loss Function for Probability Distributions" the authors propose a loss function for training against instances in which ordering within the data vector is unimportant. I do not find the proposed loss function to be well motivated, find a number of confusing points (errors?) in the manuscript, and do not easily follow what was done in the examples. First, it should be noted that this is a very restricted consideration of what it means to compare two sets since only sets of equal size are under consideration; this is fundamentally different to the ambitions of e.g. the Hausdorff measure as used in analysis. The logsumexp formulation of the proposed measure is unsatisfactory to me as it directly averages over each of the independent probabilities that a given element is a member of the target set, rather than integrating over the combinatorial set of probabilities for each set of complete possible matches. Moreover, the loss function H() is not necessarily representative of a generative distribution. The definition of the Hausdorff distance given is directional and is therefore not a metric, contrary to what is stated on page 2. I find the description of the problem domain confusing on page 3: the space [0,1]^NxF is described as binary, but then values of log y_i and log (1-y_i) are computed with y in [0,1] so we must imagine these are in fact elements in the open set of reals: (0,1). Clarity of the examples could be greatly improved, in particular by explaining precisely what is the objective of each task and what are the *ingredients* we begin with.