The Rayleigh Quotient and Contrastive Principal Component Analysis II

This paper extends contrastive principal component analysis by introducing kernel-weighted spatial (k-ρPCA) and functional (f-ρPCA) methods that unify spatial and functional data analysis within a single mathematical framework, demonstrated through applications in genomics.

The Gist

Imagine you are a detective trying to find a specific suspect in a crowded room. The room is filled with people (data points), and everyone is moving around, talking, and shifting positions. Your goal is to find the one person who is doing something unique, while ignoring the general "noise" of the crowd.

This is exactly what Contrastive PCA does for scientists analyzing biological data. It's a mathematical tool that helps find the "signal" (the interesting changes) by subtracting out the "noise" (the background variations).

This paper introduces two new, super-powered versions of this detective tool: k-ρPCA and f-ρPCA. Here is how they work, explained with simple analogies.

The Problem: Finding the Needle in the Haystack

Standard data analysis (like regular PCA) is like looking at a photo of a crowded party and trying to find the person wearing a red hat. It looks at everyone and finds the biggest differences. But often, the biggest differences are just people moving around the room, not the person in the red hat.

Contrastive PCA is smarter. It says: "I have a Target group (the party) and a Background group (a photo of the same room, but empty or with normal people). I want to find what makes the Target group different from the Background, while ignoring what they have in common."

The authors of this paper took this idea and gave it two new superpowers.

---

Superpower 1: k-ρPCA (The "Map" Detective)

The Challenge: Sometimes, data isn't just a list of numbers; it has a location. Imagine you have a map of a city where every house has a sensor measuring pollution. You want to find pollution spikes caused by a factory (Target), but you need to ignore the natural wind patterns that affect the whole city (Background).

The Analogy:
Think of k-ρPCA as a detective who carries a magnifying glass that only looks at neighbors.

• In standard analysis, the detective looks at the whole city at once.
• In k-ρPCA, the detective uses a "kernel" (a fancy word for a rule) that says, "If two houses are right next to each other, their data is connected."
• This allows the tool to spot patterns that are spatially specific. It can say, "Ah! The pollution is spiking right here in this neighborhood, which is different from the general wind patterns elsewhere."

Real-World Example in the Paper:
The team used this on colorectal cancer tissue.

• Target: A map of a tumor (where every spot on the map has gene data).
• Background: Normal tissue from a different patient (no map, just a list of genes).
• Result: The tool found genes that were acting strangely specifically in the tumor's geography, ignoring the fact that normal cells also have those genes. It successfully highlighted the "tumor zones" without needing to know exactly what every cell type was beforehand.

---

Superpower 2: f-ρPCA (The "Movie" Detective)

The Challenge: Sometimes, data isn't a snapshot; it's a movie. Imagine tracking a patient's immune system every day for two weeks after a vaccine. You have a curve of data for every person. You want to find how the "Booster" shot changes the reaction compared to the "Primer" shot.

The Analogy:
Think of f-ρPCA as a detective who watches two movies side-by-side.

• Movie A (Background): The immune response to the first vaccine dose.
• Movie B (Target): The immune response to the second (booster) dose.
• Instead of comparing day-by-day (which is messy), this tool turns the movies into smooth, continuous flowing ribbons.
• It then asks: "Where does the ribbon for the Booster wiggle or spike in a way the Primer ribbon never does?"

Real-World Example in the Paper:
They analyzed blood samples from people getting COVID-19 vaccines.

• They compared the immune response to the first dose (Primer) vs. the second dose (Booster).
• Result: The tool found that the immune system reacted sharper and faster to the second dose. Specifically, it identified genes that spiked on Day 1 for the booster, whereas the first dose took until Day 2 to peak. This gave a clear, mathematical proof of how the body "remembers" the vaccine.

---

Why This Matters (The "So What?")

Before this paper, scientists had to use different tools for maps (spatial data) and movies (time-series data). It was like having a screwdriver for screws and a hammer for nails, but no tool that could do both if the job got complicated.

This paper unifies them. It shows that whether you are looking at where things happen (space) or when things happen (time), you can use the same mathematical "Rayleigh Quotient" engine to find the unique differences.

In a nutshell:

• Old Way: "Here is a list of differences. Good luck figuring out which ones matter."
• New Way (k-ρPCA & f-ρPCA): "Here is exactly what is unique about your Target data, whether it's a specific location on a map or a specific moment in time, while ignoring all the boring background noise."

This helps doctors and biologists find the true "smoking gun" in complex diseases like cancer or vaccine responses, leading to better treatments and understanding.

Technical Summary

1. Problem Statement

Contrastive Principal Component Analysis (cPCA) is a dimensionality reduction technique designed to maximize variance in a "target" dataset while minimizing variance in a "background" dataset. While previous work established that cPCA can be formulated as a generalized eigenvalue problem maximizing a Rayleigh quotient (termed $\rho$PCA), existing methods faced limitations in two specific domains:

• Spatial Data: Standard cPCA does not inherently account for spatial correlations. Existing spatial PCA methods often treat spatial structure as incidental to total variance rather than a primary signal to be contrasted against non-spatial noise.
• Functional Data: Functional PCA (fPCA) reduces dimensionality for data represented as curves (e.g., time-series). Previous attempts at contrastive fPCA involved subtracting background covariance from target covariance. This approach suffers from mathematical instability, as the difference of two positive semi-definite matrices is not guaranteed to be positive semi-definite, potentially leading to arbitrary or non-meaningful eigenfunctions.

The authors aim to extend the $\rho$PCA framework to handle spatially structured data and functional data within a unified mathematical framework based on the Rayleigh quotient.

2. Methodology

The paper introduces two extensions to the standard $\rho$PCA method: k-$\rho$PCA (Kernel-weighted) and f-$\rho$PCA (Functional).

A. k-$\rho$PCA (Spatial Contrastive PCA)

• Concept: This method incorporates spatial structure by replacing the standard sample covariance matrix with a kernel-weighted covariance matrix.
• Mechanism:
  • A kernel matrix $K$ is constructed based on the spatial distance between observations (e.g., spots in a tissue sample). A common choice is the Gaussian kernel: $K_{ij} = \exp(-\|s_i - s_j\|^2 / 2h^2)$.
  • The target covariance matrix is redefined as $\hat{\Sigma}_K = \frac{1}{n-1} X^\top K X$, where $X$ is the normalized gene expression matrix.
  • The background covariance $\hat{\Sigma}_B$ remains standard (calculated from non-spatial data).
  • The method solves the generalized eigenvalue problem: $\arg\max_v \frac{v^\top \hat{\Sigma}_K v}{v^\top \hat{\Sigma}_B v}$.
• Advantage: This up-weights local covariance, forcing the resulting components to reflect spatial structure specifically, while the contrastive term filters out variation common to the non-spatial background.

B. f-$\rho$PCA (Functional Contrastive PCA)

• Concept: This method performs contrastive analysis directly in the space of basis function coefficients rather than on discrete time points or by subtracting covariance functions.
• Mechanism:
  • Observations (target and background) are modeled as square-integrable stochastic processes $X(t)$ and $Y(t)$.
  • Instead of discrete PCA, samples are fitted to a set of basis functions (e.g., B-splines, Fourier): $X_i(t) = \sum a_{id} b_d(t)$.
  • The data is represented by coefficient matrices $A_X$ and $A_Y$.
  • To account for the potential non-orthogonality of basis functions, a Gram matrix $G$ (where $G_{kl} = \int b_k(t)b_l(t)dt$) is used.
  • The Rayleigh quotient is optimized in the coefficient space:
    $$\arg\max_w \frac{w^\top G^{1/2} A_X^\top A_X G^{1/2} w}{w^\top G^{1/2} A_Y^\top A_Y G^{1/2} w}$$
  • The resulting eigenvectors $w$ are converted back to the function space to yield eigenfunctions $\phi(t)$.
• Advantage: This avoids the instability of subtracting covariance operators. It naturally produces smooth, interpretable eigenfunctions that represent modes of variation distinct to the target process.

3. Key Contributions

1. Unification of Frameworks: The paper demonstrates that spatial PCA, functional PCA, and contrastive PCA can all be unified under the single mathematical formalism of the Rayleigh quotient maximization via generalized eigenvalue problems.
2. k-$\rho$PCA: A novel method to identify spatially variable features in target data while controlling for cell-type variation found in non-spatial background data.
3. f-$\rho$PCA: A robust alternative to previous contrastive fPCA methods that avoids the "difference of PSD matrices" problem by operating in the basis coefficient space.
4. Theoretical Stability: Both extensions inherit the theoretical guarantees of $\rho$PCA, ensuring that the resulting directions are well-defined and maximize the target-to-background variance ratio.

4. Results and Applications

Application 1: Spatial Transcriptomics (k-$\rho$PCA)

• Dataset: Colorectal cancer (CRC) tissue samples profiled with Visium V2 and Visium HD (target) vs. non-tumor adjacent tissue scRNA-seq (background).
• Findings:
  • The first generalized eigenvector (GE1) successfully distinguished tumor spots from non-tumor spots in both Visium datasets, whereas standard PCA failed to separate them in the dominant direction.
  • GE1 identified genes with high spatial variance specific to the tumor, including ASCL2, EREG, and SFRP (associated with prognosis and metastasis).
  • GE2 highlighted a fibroblast response in the tumor core involving ITGBL1 and SFRP4.
  • Significance: The method identified tumor-specific mechanisms without requiring prior cell-type annotation or matched spatial background samples, leveraging public non-spatial scRNA-seq data as a background.

Application 2: Longitudinal Gene Expression (f-$\rho$PCA)

• Dataset: Longitudinal bulk RNA-seq data from 23 subjects receiving two doses of an mRNA vaccine (Primer vs. Booster).
• Findings:
  • The method contrasted the immune response of the "Booster" (target) against the "Primer" (background).
  • The first eigenfunction revealed that the interferon response in the booster dose was "sharper" and peaked on Day 1, whereas the primer response peaked on Day 2.
  • Top genes identified (GBP2, ISG20, SP110, LAP3) showed significantly higher variance in the booster response, consistent with known SARS-CoV-2 biology.
  • Significance: f-$\rho$PCA allowed for a joint analysis of two time courses, identifying differential temporal dynamics that might be missed by separate post-hoc comparisons.

5. Significance

• Methodological Advancement: By extending $\rho$PCA to spatial and functional domains, the authors provide a flexible toolkit for modern high-dimensional biological data, which is increasingly spatial (spatial transcriptomics) and temporal (longitudinal omics).
• Biological Insight: The methods successfully extracted biologically relevant signals (tumor microenvironment dynamics and vaccine response kinetics) that were obscured by standard variance-based methods.
• Practical Utility: The approach allows researchers to use publicly available, non-matched background datasets (e.g., standard scRNA-seq) to analyze complex spatial or functional target data, lowering the barrier for high-quality contrastive analysis.
• Computational Efficiency: Since the core solution relies on standard generalized eigenvalue solvers, the methods are scalable to large datasets (e.g., Visium HD with hundreds of thousands of bins).

In summary, this paper successfully generalizes the Rayleigh quotient framework to create robust, unified methods for contrastive analysis of spatial and functional data, offering a mathematically sound alternative to ad-hoc subtraction methods in functional data analysis.