The definition of Linear Symmetry-Based Disentanglement (LSBD) formalizes the notion of linearly disentangled representations, but there is currently no metric to quantify LSBD. Such a metric is crucial to evaluate LSBD methods and to compare to previous understandings of disentanglement. We propose $\mathcal{D}_\mathrm{LSBD}$, a mathematically sound metric to quantify LSBD, and provide a practical implementation for $\mathrm{SO}(2)$ groups. Furthermore, from this metric we derive LSBD-VAE, a semi-supervised method to learn LSBD representations. We demonstrate the utility of our metric by showing that (1) common VAE-based disentanglement methods don't learn LSBD representations, (2) LSBD-VAE as well as other recent methods can learn LSBD representations, needing only limited supervision on transformations, and (3) various desirable properties expressed by existing disentanglement metrics are also achieved by LSBD representations.

翻译：线性对称分解(LSBD)的定义正式确定了线性分解(LSBD)的概念,但目前没有量化LSBD的衡量标准。这种衡量标准对于评估LSBD的方法和比较先前对分解的理解至关重要。我们建议使用一个数学上健全的衡量标准,用于量化LSBD,并为美元/mathrm{SO}(2)美元群体提供实际实施。此外,我们从这一指标中得出LSBD-VAE,这是学习LSBD代表的半监督方法。我们通过表明(1)基于VAE的共性分解方法不会学习LSBD的衡量方法,(2)LSBD-VAE以及其他最近的方法,表明我们的指标的效用,表明LSBD还能够了解LSBD的表述,只需对变异性进行有限的监督,以及(3)现有分解指标表达的各种可取的特性。