Обсуждение

If you use dim. reduction, you may be interested in two recent preprints we've posted on contrastive PCA: The Rayleigh Quotient and Contrastive Principal Component Analysis I & II w/ Maria Carilli & Kayla Jackson. They cover a lot of ground from theory to practice. 1/🧵

Lior Pachter

13 апр., 02:48

Part I explains why the contrastive PCA of @jameszou.bsky.social is flawed, and (at best) can be viewed as a Taylor approximation of a Rayleigh quotient. Optimization of the RQ directly is not only possible but tractable via a generalized eigenvalue problem. 2/ www.biorxiv.org/content/10.1...

www.biorxiv.org

Lior Pachter

13 апр., 02:48

The Rayleigh quotient for two square matrices A and B is defined as R(v) = (vᵀAv) / (vᵀBv). Maximization of this RQ, which is easy to solve via the (generalized) eigenvalue problem Av = 𝜆Bv. This is illustrated below (a) target and bg, (b) contrastive PCA (c) variance. 3/

Lior Pachter

13 апр., 02:48

Whereas @jameszou.bsky.social's approach to contrastive PCA has a parameter alpha (that can be difficult / impossible to choose), maximizing the RQ (we call this ρPCA) is straightforward, and produced much better results. We show this with many examples. 4/

Lior Pachter

13 апр., 02:48

ρPCA is super useful in anytime one wants to examine differences in variance (rather than just the mean), while performing dim. reduction. Our preprint gives an example using single-cell RNA-seq: we find differentially variable genes (rather than differentially expressed). 5/

Lior Pachter

13 апр., 02:48

The fundamental flaw with @jameszou.bsky.social's heuristic contrastive PCA is a subtraction of covariance matrices (which many not be +ve semidefinite!). Even worse is differential robust PCA (drPCA) by Eric Xing which subtracts data matrices 😱. Unfortunately Var(X-Y) = Var(X+Y)... 6/

Lior Pachter

13 апр., 02:48

We found the field as a whole (contrastive PCA) to consist of heuristics layered on heuristics, and in this case it turns out doing the math right really matters. There are of course non-linear contrastive methods but ρPCA is interpretable which can matter a lot 7/

Lior Pachter

13 апр., 02:48

The ρPCA method is implemented in an easy to use Python package github.com/pachterlab/r... Quick to install. Fast. We now use it routinely whenever engaging in exploratory data analysis. 8/

GitHub - pachterlab/rhopcaContribute to pachterlab/rhopca development by creating an account on GitHub.github.com

Lior Pachter

13 апр., 02:48

In Part II, we develop ρPCA in two ways: adding kernel weighting (from spatial PCA), and also operating in the space of basis coefficients (from functional PCA). The former is great for spatial analysis, the latter for time series. 9/