Disentangling Interpretable Factors with Supervised Independent Subspace Principal Component Analysis

Contrastive representation learning, as introduced in Abid et al. [2018], aims to find representations that capture the difference between explicit positive and negative samples. For example, contrastive PCA (cPCA) takes the pair of a target dataset of interest and a background dataset as inputs and finds the principal components that account for large variances in the target but small variances in the background [Abid et al., 2018]. cVAE [Abid and Zou, 2019] and contrastiveVI [Weinberger et al., 2023] apply the idea to VAE architectures, where the target and background data are assumed to have different generative processes and the corresponding representations lie in separate subspaces. The learning of the target and background subspaces are achieved by explicitly forcing target and background samples to use different spaces. More recently, Tu et al. [2024] and Qiu et al. [2023] adopt an HSIC-based regularization approach to encourage the disentanglement of the two subspaces.

Despite the similarity, our proposed sisPCA model is conceptually different from these contrastive models. sisPCA (and the non-linear HCV [Lopez et al., 2018]) focuses on the decomposition of a single dataset, as opposed to a pair of target and background datasets. The sense of contrast in sisPCA comes from explicit supervision, of which each subspace either aligns with or is repulsed from. If we concatenate the pair of target-background into one dataset and consider the case-control information as supervision, the associated sisPCA subspace can also reflect the difference similar to cPCA (the objective is still not the same).

We first provide a motivating example to see how the first supervision term in (2) affects the optimization landscape. Assume no supervision $Y$ (i.e. $K_{Y_{m}}=I$) and all the latent subspaces $\{Z_{m}\}$ share the same dimension $d$. In the absence of disentanglement regularization, every subspace will simply be the same vanilla PCA space, leading to complete overlap. The introduction of the disentanglement penalty in (2) maintains the objective’s symmetry with respect to ${U_{m}}$. In other words, subspaces are interchangeable in the pure unsupervised setting, and consequently the symmetry gives rise to multiple local optima. Nevertheless, the presented challenge is trivial, in that all local optima are also global, and the number of local optima, considered in terms of invariant groups up to column sign-flipping, depends only on the number of subspaces $m$.

Under supervision, some subspaces may be selectively favored. For example, the scaling of subspace supervision kernels $\{K_{Y_{m}}\}$ can break the symmetry of the overall objective, allowing information to be preferentially preserved in one subspace while being removed from others. In such scenarios (referred to as unbalanced supervision), local optima that are closer to the supervised PCA results of subspaces with larger $K_{Y_{m}}$ also yield higher objective values. The balanced supervision condition can be evaluated by calculating the relative ratio of kernel scales, or equivalently, the ratio of objective gradient with respect to each subspace kernel.

Another important observation is that, because the disentanglement penalty is symmetric, tuning the hyperparameter $\lambda$ will not alter the ranking of local optima. This suggests a straightforward initialization strategy to navigate towards the global solution by evaluating the relative supervision strength, $HSIC(Z_{m}^{0},Y_{m})$, for each subspace. Here, $Z_{m}^{0}$ represents the initial, pre-disentanglement solution for subspace $m$, which can be efficiently computed using supervised PCA.

We can directly visualize the optimization landscape in a simplified scenario (Fig. 7). Consider $X=\begin{pmatrix}1&0\\ -1&0\end{pmatrix}$, which consists of two 2D vectors separated along the first axis. With supervisions $Y_{1}=(1,-1)^{T}$ and $Y_{2}=cY_{1}$ differing only by a scalar $c$, our goal is to determine the projections $U=(u,u_{2})^{T}$ and $V=(v,v_{2})^{T}$ that transform $X$ into subspace representations $Z_{1}=XU=(u,-u)^{T}$ and $Z_{2}=XV=(v,-v)^{T}$. Adopting a linear kernel for $K_{Y}$, the sisPCA-linear objective (2) is now equivalent to the following simplified problem

Since $X$ only has one effective dimension, both $U$ and $V$ align with that axis (i.e., $u=1$ and $v=1$, the unregularized solution) when the disentanglement penalty $\lambda$ is low ($\lambda<1$), leading to a single optimal solution $(u,v)=(1,1)$. Under strong regularization, two local maxima $(1,0)$ and $(0,1)$ emerge, and their relative order depends solely on the scalar $c$ and is unaffected by $\lambda$. Specifically, each maximum represents a preference for capturing the significant axis in $X$ within one subspace, allowing for an easy decision on initialization based on a comparison between $u$ and $cv$. In the context of alternating optimization methods like Algorithm 1, it can be simply done by fixing the subspace $V$ ($v=1$) and updating $U$ first to minimize the disentanglement penalty (from $u=1$, the unregularized solution, to $u=0$). The optimization path stays near the global optima regardless of $\lambda$.

This phenomenon also resembles the situation described in Ge et al. [2017] on $f(UV^{T})$, where an additional regularizer $||U^{T}U-V^{T}V||_{F}^{2}$ is necessary to maintain balance when dealing with asymmetric matrices. The first half of Conjecture 3.1 is indeed motivated by the above result. For sisPCA, a similar regularizer in the form of $||Z_{i}Z_{i}^{T}-Z_{i}Z_{i}^{T}||_{F}^{2}$ may be enough to ensure balanced supervision. Moreover, given that the multi-solution challenge arises from subspace interchangeability, we hypothesize that more discriminative $Y$ could alleviate the issue. Since there are only pairwise interactions between subspaces in (2), Ge et al. [2017]’s framework can be adapted to study the behavior of more than two subspaces. Intuitively, the proof of Conjecture 3.1 would be to show that the local gradient direction of (2) aligns with the global descent direction in a region near global optima, and proper regularization or initialization can make sure the optimizer stays in that region. Following the procedures described in section 5 of Ge et al. [2017] for asymmetric matrices, we may concatenate all subspaces into $W:=[Z_{1},...,Z_{m}]^{T}$ and reformulate the objective as a quadratic function of $N:=WW^{T}$ plus some regularization.

In this section, we discuss the sisPCA-general optimization problem in detail. Recall that it is motivated by the limitation of sisPCA-linear that zero HSIC disentanglement regularization does not ensure statistical independence. In sisPCA-general, we want to solve (1) by employing HSIC with with universal kernels, e.g. the Gaussian kernel.

A nice result on universal kernel from the Constrained Covariance (COCO) framework [Gretton et al., 2005b] is that it can approximate any continuous function on a compact metric space. In other words, the HSIC with a universal kernel will be zero only if, among all possible continuous transformation, the covariance of two random variables are always zero. In sisPCA-general, the latent representation is still encoded by a linear projection $Z_{j}=XU_{j}$. This indicates that $U_{j}$ can be interpreted in the same way as PCA and sisPCA-linear to extract feature importance.

We solve the sisPCA-general problem with Gaussian kernel,

using gradient descent (Algorithm 2). Since the problem is constrained, the algorithm unfortunately is not guaranteed to converge. The HSIC with Gaussian kernel also affects the optimization landscape. Empirically, we observed that sisPCA-general tends to have many local optima, requiring exhaustive tuning of training parameters such as the learning rate and stopping criteria. The Gaussian kernel scale $w$ is also important. Ma et al. [2020] reported in their experiments that the performance of HSIC-based optimization is moderately sensitive to $w$ and thereby proposed a multiple-scale solution that essentially aggregates results from different $w$. In our experiments, we followed Gretton et al. [2005a] by setting the kernel scale to the median distance between data points in the input space.

Another notable disadvantage of sisPCA-general is its cubic-scaling computational complexity. Here we focus on scenarios where the number of features $p$ is fixed and the sample size $n$ is large ($n>p$), which is common when data preprocessing and filtering is in place. Specifically, the computational complexity for calculating $HSIC(X,Y)$ with a linear kernel for $X\in\mathbb{R}^{n\times p}$ and a given target kernel $K_{Y}$ for $Y$ is $O(n^{2}p)$. If both $K_{X}$ and $K_{Y}$ are linear, the complexity can be further reduced to $O(np)$. Additionally, the eigendecomposition of the (cross) covariance incurs a cost of $O(p^{3})$. This means that each update in Algorithm 1 has an overall complexity of $O(n^{2})$ wrt the sample size $n$, aligning with that of supervised PCA. In comparison, most dimensionality reduction methods requiring pair-wise distance computation have a complexity of at least $O(n^{2})$. On the other hand, for sisPCA-general, computing a Gaussian-kernel $HSIC(X,Y)$ naively is of $O(n^{3})$ complexity. From an engineering perspective, further optimizations for both the sisPCA-linear (Algorithm 1) and sisPCA-general (Algorithm 2) algorithms are conceivable, for instance, by exploiting data sparsity and by mini-batch or incremental update.

Following kernel PCA’s methodology [Schölkopf et al., 1997], we can extend sisPCA for nonlinear scenarios where features in $X$ interact through a feature map $\Phi:\mathbb{R}^{p}\rightarrow F$. The space $F$ may have infinite dimension, and we can only explicitly write out the projection $U$ in the finite case $F\subset\mathbb{R}^{k}$, for example, $\Phi(x)=x^{2}$. In that context, the latent representation $Z:=\Phi(X)U$ can be naively computed, and the sisPCA objective (1) will stay the same. By simply replacing $X$ with $\Phi(X)$, methods such as gradient descent and Algorithm 1 still hold. However, the beauty of the kernel trick is that we can bypass the computation of $\Phi(X)$ to obtain $Z$. Instead of learning the projection matrix $U$, which is no longer possible when $F$ is infinite-dimensional, we can learn a set of coefficients $\vec{\alpha}$ from

with $\{X_{i}\}$ representing the $n$ data points and $k(x,y):=\langle\Phi(x),\Phi(y)\rangle$ the corresponding kernel of $\Phi$. Note that $k(\cdot,\cdot)$ is specified wrt to the data $X$, which is different from the kernel $K_{z}$ for latent space and the kernel $K_{y}$ for supervision. We leave the practical implementation of this extension to future exploration.

We adapt the approach from Abid et al. [2018] to develop a computational pipeline for automated selection of the sisPCA hyperparameter $\lambda$. The process involves:

1. 1.

   Fitting sisPCA models with multiple $\lambda$ values.

2. 2.

   Computing pairwise similarity (affinity) between learned subspaces.

3. 3.

   Identifying representative subspace clusters using spectral clustering.

We compute subspace affinity based on the principal angle [Miao and Ben-Israel, 1992]:

where $Z\in\mathcal{R}^{n\times k}$ is the orthonormal basis of a dim-k subspace with $n$ observations, and the principal angles $\Theta$ are computed from the SVD of $Z_{1}^{T}Z_{2}=U\cos(\Theta)V^{T}$.

Through empirical testing, we found the Fubini-Study Grassmann distance provides optimal performance. Thus, we define subspace affinity as:

For stability, we truncate subspace latent dimension to $k^{\prime}:=\min(\text{rank}(K_{Y}),k)$, where $K_{Y}$ is the corresponding supervision kernel (see discussion on subspace effective dimension in Appendix F.3). For sisPCA models with multiple subspaces, we compute model-wise affinity as the average of subspace affinities

For completeness, here we provide subspace visualization for baseline models not included in the main figures. Specifically, Fig. 11 shows the full sPCA performance on the simulated data, related to Fig. 3. Fig. 12 visualizes the single-cell subspaces learned by sPCA, VAE, supVAE, and hsVAE-sc, related to Fig. 5.

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4. Justification: In the paper, we proposed a multi-subspace extension of PCA to disentangle interpretable subspaces of variations using self-supervision. We believe our work is highly novel and significant to practitioners using PCA as well as general audience in computational biology.

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14. Justification: The proof of Remark 3.2 is stated following the remark in the main text. The proofs of Remark 3.1 and 3.3 can be found in Appendix B. Detailed discussion on 3.1 can be found in the Appendix C.

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