Deep metric learning (DML) aims to automatically construct task-specific
distances or similarities of data, resulting in a low-dimensional
representation. Several significant metric-learning methods have been proposed.
Nonetheless, no approach guarantees the preservation of the ordinal nature of
the original data in a low-dimensional space. Ordinal data are ubiquitous in
real-world problems, such as the severity of symptoms in biomedical cases,
production quality in manufacturing, rating level in businesses, and aging
level in face recognition. This study proposes a novel angular triangle
distance (ATD) and ordinal triplet network (OTD) to obtain an accurate and
meaningful embedding space representation for ordinal data. The ATD projects
the ordinal relation of data in the angular space, whereas the OTD learns its
ordinal projection. We also demonstrated that our new distance measure
satisfies the distance metric properties mathematically. The proposed method
was assessed using real-world data with an ordinal nature, such as biomedical,
facial, and hand-gestured images. Extensive experiments have been conducted,
and the results show that our proposed method not only semantically preserves
the ordinal nature but is also more accurate than existing DML models.
Moreover, we also demonstrate that our proposed method outperforms the
state-of-the-art ordinal metric learning method.

本研究提出了一种基于角三角形距离和有序三元组网络的新深度度量学习方法，能够准确地将有序数据嵌入一个低维空间表示。实验结果表明，该方法不仅能保持有序性，而且比现有的深度度量学习模型更准确。

基于序缩放学习的角三角距离

Angular triangle distance for ordinal metric learning

Generative flows and diffusion models have been predominantly trained on
ordinal data, for example natural images. This paper introduces two extensions
of flows and diffusion for categorical data such as language or image
segmentation: Argmax Flows and Multinomial Diffusion. Argmax Flows are defined
by a composition of a continuous distribution (such as a normalizing flow), and
an argmax function. To optimize this model, we learn a probabilistic inverse
for the argmax that lifts the categorical data to a continuous space.
Multinomial Diffusion gradually adds categorical noise in a diffusion process,
for which the generative denoising process is learned. We demonstrate that our
method outperforms existing dequantization approaches on text modelling and
modelling on image segmentation maps in log-likelihood.

本文介绍了生成流和扩散模型的扩展，用于类别数据，例如语言或图像分割：Argmax 流和多项式扩散，并证明了该方法在文本建模和图像分割映射建模上优于现有的量化去噪方法。

Argmax 流和多项式扩散：学习分类分布

Argmax Flows and Multinomial Diffusion: Learning Categorical  Distributions

We introduce a new non-negative matrix factorization (NMF) method for ordinal
data, called OrdNMF. Ordinal data are categorical data which exhibit a natural
ordering between the categories. In particular, they can be found in
recommender systems, either with explicit data (such as ratings) or implicit
data (such as quantized play counts). OrdNMF is a probabilistic latent factor
model that generalizes Bernoulli-Poisson factorization (BePoF) and Poisson
factorization (PF) applied to binarized data. Contrary to these methods, OrdNMF
circumvents binarization and can exploit a more informative representation of
the data. We design an efficient variational algorithm based on a suitable
model augmentation and related to variational PF. In particular, our algorithm
preserves the scalability of PF and can be applied to huge sparse datasets. We
report recommendation experiments on explicit and implicit datasets, and show
that OrdNMF outperforms BePoF and PF applied to binarized data.

本文介绍了一种新的非负矩阵分解方法 OrdNMF，可以在推荐系统中处理分类数据，而不需要进行二值化处理，并且在数据表示和算法效率方面都具有优势。

Ordinal Non-negative Matrix Factorization for Recommendation

Motivated by generating personalized recommendations using ordinal (or
preference) data, we study the question of learning a mixture of MultiNomial
Logit (MNL) model, a parameterized class of distributions over permutations,
from partial ordinal or preference data (e.g. pair-wise comparisons). Despite
its long standing importance across disciplines including social choice,
operations research and revenue management, little is known about this
question. In case of single MNL models (no mixture), computationally and
statistically tractable learning from pair-wise comparisons is feasible.
However, even learning mixture with two MNL components is infeasible in
general.
Given this state of affairs, we seek conditions under which it is feasible to
learn the mixture model in both computationally and statistically efficient
manner. We present a sufficient condition as well as an efficient algorithm for
learning mixed MNL models from partial preferences/comparisons data. In
particular, a mixture of $r$ MNL components over $n$ objects can be learnt
using samples whose size scales polynomially in $n$ and $r$ (concretely,
$r^{3.5}n^3(log n)^4$, with $r\ll n^{2/7}$ when the model parameters are
sufficiently incoherent). The algorithm has two phases: first, learn the
pair-wise marginals for each component using tensor decomposition; second,
learn the model parameters for each component using Rank Centrality introduced
by Negahban et al. In the process of proving these results, we obtain a
generalization of existing analysis for tensor decomposition to a more
realistic regime where only partial information about each sample is available.

该论文研究了使用偏序（或偏好）数据生成个性化推荐的方法，通过学习 MultiNomial Logit（MNL）模型组合的问题，提出了一种有效的算法来从部分有序或偏好数据中学习混合的 MNL 模型，从而可生成更准确的推荐结果。