Improve Representation for Imbalanced Regression through Geometric Constraints (2503.00876v1)

Abstract: In representation learning, uniformity refers to the uniform feature distribution in the latent space (i.e., unit hypersphere). Previous work has shown that improving uniformity contributes to the learning of under-represented classes. However, most of the previous work focused on classification; the representation space of imbalanced regression remains unexplored. Classification-based methods are not suitable for regression tasks because they cluster features into distinct groups without considering the continuous and ordered nature essential for regression. In a geometric aspect, we uniquely focus on ensuring uniformity in the latent space for imbalanced regression through two key losses: enveloping and homogeneity. The enveloping loss encourages the induced trace to uniformly occupy the surface of a hypersphere, while the homogeneity loss ensures smoothness, with representations evenly spaced at consistent intervals. Our method integrates these geometric principles into the data representations via a Surrogate-driven Representation Learning (SRL) framework. Experiments with real-world regression and operator learning tasks highlight the importance of uniformity in imbalanced regression and validate the efficacy of our geometry-based loss functions.

• The paper proposes two novel geometric constraints, enveloping and homogeneity losses, to learn better representations for imbalanced regression datasets.
• These losses encourage the learned representations to cover the feature space uniformly and distribute evenly along the data trace.
• The Surrogate-driven Representation Learning (SRL) framework is introduced to apply these geometric losses effectively to mini-batches by using a surrogate of centroids representing all target bins.

This paper addresses the problem of imbalanced regression, where the goal is to train a model on a dataset where certain target values are much more frequent than others. This is a common problem in many real-world applications, such as age estimation from faces or predicting the similarity between sentences.

The paper argues that previous work has focused mainly on classification tasks and that these methods are not directly applicable to regression because they don't account for the continuous and ordered nature of the target values in regression. The paper identifies that a key problem in imbalanced regression is that the learned representations of the data are not uniformly distributed in the feature space. This means that the model focuses on the frequent target values and neglects the rare ones.

To address this, the paper introduces two new loss functions that encourage the learned representations to be more uniformly distributed:

• Enveloping loss: This loss encourages the "trace" of the data representations (the path that the representations follow as the target value changes) to cover the entire feature space.
• Homogeneity loss: This loss ensures that the data representations are evenly spaced along the trace, so that the model doesn't focus on certain regions of the feature space more than others.

These two losses act as geometric constraints on the latent trace. The problem is that these losses cannot be applied to representations from a single mini-batch, because a mini-batch is unlikely to cover the full range of labels. To address this, the paper introduces a Surrogate-driven Representation Learning (SRL) framework.

Here's how Surrogate-driven Representation Learning (SRL) works: 1. The representations of the same bins within a mini-batch are averaged to form centroids.

1. Missing bins are "re-filled" by taking corresponding centroids from the previous epoch.
2. The geometric losses are then applied to this surrogate, which contains centroids for all bins.

The Surrogate-driven Representation Learning (SRL) framework is trained end-to-end with a combination of the mean squared error (MSE) loss, the geometric losses, and a contrastive loss. The contrastive loss encourages the representations of data points to be close to the centroids of their corresponding bins and far from the centroids of other bins.

The paper also introduces a new benchmark for imbalanced regression called Imbalanced Operator Learning (IOL). This task involves training models on imbalanced domain locations in function space mapping.

In summary, the contributions of the paper are:

• Two novel loss functions, the enveloping loss and the homogeneity loss, to encourage uniform feature distribution for imbalanced regression.
• The Surrogate-driven Representation Learning (SRL) framework that incorporates these geometric principles into data representations.
• The Imbalanced Operator Learning (IOL) task, which is a new benchmark for imbalanced regression.
• Experiments on real-world regression and operator learning tasks that validate the effectiveness of the proposed method.

Preliminaries

The method relies on some math. Let's walk through it. The paper uses $\mathbf{x}_i$ to represent the input data and $y_i$ to represent the corresponding continuous target value.

• $\mathbf{x}_i$ is the input.
• $y_i$ is the continuous target value for the input.

A neural network $f(\cdot)$ is used to generate a feature representation $\mathbf{z}_i$ of $\mathbf{x}_i$.

• $f(\cdot)$ is a neural network.
• $\mathbf{z}_i = f(\mathbf{x}_i)$ is the feature representation of $\mathbf{x}_i$ generated by the neural network.

The feature representation is normalized so that its length is 1 ( $\left\|\mathbf{z}_i\right\|=1$ ). This ensures that all feature representations lie on the surface of a unit hypersphere.

The dataset is divided into $K$ unique bins. A surrogate is defined as a set of centroids $\mathbf c_k$, where each centroid represents a distinct bin. These centroids are computed by averaging the representations $\mathbf{z}$ that share the same bin, and they are also normalized to have a length of 1 ($\left\|\mathbf{c}_k\right\|=1$).

• $K$ is the number of unique bins in the dataset.
• $\mathbf c_k$ is the centroid for bin $k$.

A path $l$ is defined as a continuous curve that maps each target value $y_k$ to its corresponding centroid $\mathbf c_k$. In math terms, $l:[y_{\mathrm{min},y_{\mathrm{max}] \mapsto \mathbb{R}^n$ with $\left\|l(y)\right\|=1$, such that $l( y_k) = \mathbf c_k$.

• $l$ is a continuous curve that maps each target value $y_k$ to its corresponding centroid $\mathbf c_k$.

Enveloping Loss

To encourage the trace of regression representations to fill the entire unit hypersphere, the paper introduces the concept of enveloping loss. This loss is inspired by the analogy of wrapping yarn around a ball, where the yarn represents the latent trace and the ball represents the feature space. The goal is to maximize the hypervolume of a tubular neighborhood around the latent trace, relative to the total hypervolume of the hypersphere.

The tubular neighborhood $T(l,\epsilon)$ of $l$ is defined as the set of all unit vectors $\mathbf{z}$ in $\mathbb{R}^n$ such that the dot product between $\mathbf{z}$ and some point $\mathbf{t}$ on the trace is greater than $\epsilon$.

$$T(l,\epsilon)=\{ \mathbf{z} \in \mathcal{U} \ | \ \mathbf{t}\cdot\mathbf{z} > \epsilon \ \text{for some} \ \mathbf{t} \in \mathrm{Im}(l)\}$$

• $\mathcal{U}$ is the set of all unit vectors in $\mathbb{R}^n$.
• $\epsilon \in (0,1)$ is a parameter that controls the size of the tubular neighborhood.
• $\mathrm{Im}(l)$ is the image of the path $l$, i.e., the set of all points on the trace.

The enveloping loss is then defined as the negative of the hypervolume of the tubular neighborhood, divided by the hypervolume of the hypersphere:

$\mathcal{L}_{\text{env} = -\frac{\text{vol}(T(l,\epsilon))}{\text{vol}(\mathcal{U})}$

• $\text{vol}(\cdot)$ returns the hypervolume of its input.

Since it is difficult to directly compute the hypervolume of the tubular neighborhood, the paper proposes a continuous-to-discrete strategy. First, $N$ points are generated that are uniformly distributed across the hypersphere. Then, the fraction of these points that fall within the neighborhood $\epsilon$ is determined. This fraction approximates the proportion of the hypersphere covered by the tubular neighborhood.

To adapt $\mathcal{L}_{\text{env}$ to discrete datasets, the optimization objective is reformulated as:

$\max \lim_{N\rightarrow\infty}\frac{P(N)}{N}$

where

$$P(N):=|\{\mathbf{p}_i \ | \ \max_{y}\{\mathbf{p}_i\cdot l(y)\}>\epsilon, i \in [N]\}|$$

• $P(N)$ is the number of points $\mathbf{p}_i$ that fall within the neighborhood $\epsilon$.
• $N$ is the total number of points that are uniformly distributed across the hypersphere.

In practice, the paper maximizes the cosine similarity between each point $\mathbf{p}_i$ and its closest point on the trace, instead of directly defining $\epsilon$. This is because the binarization required to determine if a $\mathbf{p}_i$ is within the $\epsilon$-tube is not differentiable.

Homogeneity Loss

The enveloping loss encourages the representations to cover the entire feature space, but it doesn't guarantee that the representations are evenly distributed along the trace. To address this, the paper introduces a homogeneity loss that encourages the trace to be smooth and the representations to be uniformly distributed along it.

The homogeneity loss is defined as:

$\mathcal{L}_{\text{homo}=\int_{y_{\mathrm{min}^{y_{\mathrm{max} \left\|\frac{\mathrm{d}l(y)}{\mathrm{d}y}\right\|^2 \mathrm{d}y}$

This loss penalizes the arc length of the trace. A shorter arc length indicates a smoother trace and a more uniform distribution of representations.

For discrete datasets, the homogeneity loss is defined as a summation of the squared differences between adjacent points:

$\mathcal{L}_{\text{homo} = \sum_{k=1}^{K-1} \frac{\left\|l(y_{k+1}) - l(y_k)\right\|^2}{y_{k+1}-y_{k}}$

The paper proves that with a given image of $l$, the homogeneity loss is minimized if and only if the representations are uniformly distributed along the trace.

The geometric constraints ($\mathcal{L}_\text{G}$) are then formulated as a combination of the enveloping loss and the homogeneity loss:

$\mathcal{L}_\text{G} = \lambda_{e}\mathcal{L}_{\text{env} + \lambda_{h}\mathcal{L}_{\text{homo}$

• $\lambda_{e}$ and $\lambda_{h}$ are weights that control the relative importance of the two losses.

Surrogate-Driven Representation Learning (SRL)

To apply the geometric losses to mini-batches of data, the paper introduces the Surrogate-driven Representation Learning (SRL) framework. This framework calculates the geometric loss on a surrogate instead of a mini-batch. The surrogate is a set of centroids that represent the full range of target values.

The surrogate is constructed as follows:

1. For each mini-batch, the centroids for each bin are calculated by averaging the representations of all data points in that bin.
2. Missing bins are "re-filled" by taking the corresponding centroids from the previous epoch.
3. The geometric losses are then applied to this surrogate.

To further encourage the representations of data points to be close to the centroids of their corresponding bins, the paper introduces a contrastive loss:

$\mathcal{L}_{\text{con} = -\sum_{m=1}^{M}{\log \frac{\exp(\text{sim}(\mathbf{z}_m, \mathbf c_{y}))}{\sum_{y^* \in \mathcal{Y}^*} \exp(\text{sim}(\mathbf{z}_{m}, \mathbf c_{y^*}))}$

• $\text{sim}(\cdot)$ is the cosine similarity between two inputs.

The total loss used to train the model is then:

$\mathcal{L}_{\theta}=\mathcal{L}_{\text{reg}+\mathcal{L}_{\text{G}+\mathcal{L}_{\text{con}$

• $\mathcal{L}_{\text{reg}$ is the mean squared error (MSE) loss.

Experiments

The paper evaluates the proposed method on a variety of real-world regression and operator learning tasks. The results show that the proposed method outperforms existing methods for imbalanced regression.

The paper also analyzes the impact of the different components of the proposed method. The results show that both the enveloping loss and the homogeneity loss are important for achieving good performance.

The code is available on Github.

Hopefully, this gives you a better understanding of the paper! Let me know if you have any other questions.