# Theoretical Optimizations for LD Score Regression

The computational bottleneck of Linkage Disequilibrium (LD) Score regression lies in calculating the squared Pearson correlation, $r_{jk}^2$, between a target single nucleotide polymorphism (SNP) $j$ and all other SNPs $k$ within a defined genomic window $W_j$. For a genotype matrix of $N$ individuals and $M$ SNPs, calculating the exact LD scores requires $O(N \cdot M \cdot |W|)$ operations.

Below are several theoretical and algorithmic modifications that could dramatically improve performance in a high-performance Rust implementation, drawing on techniques from numerical linear algebra, digital signal processing, and hardware-level optimization.

## 1. Dimensionality Reduction and Low-Rank Approximations

Genotype matrices exhibit high multicollinearity due to the block-like structure of haplotypes in the human genome. The effective degrees of freedom within a 1 cM window are vastly smaller than the number of SNPs.

* **Truncated Singular Value Decomposition (SVD)**: Instead of computing pairwise correlations for all SNPs in a window, project the standardized genotype matrix $X_{W}$ onto its top $K$ principal components. If $X_{W} \approx U \Sigma V^T$, the correlation between SNPs can be approximated by the dot products of the right singular vectors.

$$r_{jk} \approx \sum_{i=1}^{K} V_{ji} V_{ki}$$

```
This reduces the complexity of computing the correlation matrix from $O(N \cdot |W|^2)$ to $O(K \cdot |W|^2)$, where $K \ll N$.

```

* **Randomized Sketching**: Use Gaussian random projections or the Fast Johnson-Lindenstrauss Transform to compress the row dimension (individuals) of the genotype matrix from $N$ to a much smaller $d$. The Euclidean geometry of the columns (SNPs) is preserved, allowing for the rapid estimation of correlation coefficients with bounded error, accelerating the inner loops of the matrix multiplications.

## 2. Convolutional and Frequency-Domain Approaches

If the LD window is defined by physical distance (base pairs) or an evenly interpolated genetic map, computing sliding window cross-correlations is mathematically equivalent to a discrete convolution.

* **Fast Fourier Transform (FFT)**: By utilizing the convolution theorem, the cross-correlation of genotype vectors can be computed in the frequency domain.

$$\mathcal{F}(X * Y) = \mathcal{F}(X) \cdot \mathcal{F}(Y)^*$$

```
For highly dense SNP arrays, padding the genotype sequences and applying an FFT could theoretically reduce the $O(M \cdot |W|)$ sliding window complexity to $O(M \log M)$. This is particularly performant if the computations are offloaded to specialized hardware or aggressively parallelized across threads.

```

## 3. Hardware-Level Bit Manipulation

Raw genotype data represents biallelic SNPs in two bits (homozygous reference, heterozygous, homozygous alternate, or missing). Unpacking these into floating-point numbers for standard BLAS operations introduces immense memory bandwidth overhead.

* **SIMD Population Counts (Popcnt)**: Dot products between SNPs can be evaluated entirely in bit-space using XOR and AND operations followed by hardware population counts (`_mm512_popcnt_epi64` via AVX-512 instructions). The mathematical relationship between the raw bitwise matches and the Pearson correlation can be pre-calculated.

$$r_{jk} = \frac{E[X_j X_k] - E[X_j]E[X_k]}{\sigma_{X_j} \sigma_{X_k}}$$

```
By isolating the $E[X_j X_k]$ term, which represents the co-occurrence of alternate alleles, the algorithm only needs to count matching bits. Rust's `core::arch` or `std::simd` modules can vectorise these bitwise operations, keeping the data in L1 cache and drastically improving throughput while avoiding floating-point math entirely until the final score aggregation.

```

## 4. Stochastic Trace Estimation

The LD score $l_j$ is the sum of squared correlations. In matrix terms, the vector of LD scores for a window is the diagonal of the squared correlation matrix $R^2$.

* **Hutchinson's Trick**: Trace estimators can approximate the diagonal of a matrix without explicitly forming the entire matrix. For a local correlation matrix $R$ and a random vector $z$ with Rademacher entries (values of $+1$ or $-1$), the diagonal elements can be approximated stochastically.

$$\text{diag}(R^2) \approx \mathbb{E}[ (Rz) \odot (Rz) ]$$

```
Where $\odot$ is the Hadamard (element-wise) product. Because $R = \frac{1}{N} X^T X$, calculating $Rz$ only requires successive matrix-vector multiplications ($X^T (X z)$), which avoids the $O(|W|^2)$ matrix-matrix multiplication bottleneck entirely and operates in $O(N \cdot |W|)$.

```

## 5. Precision Reduction and Adaptive Subsampling

* **Adaptive Subsampling**: The variance of the sample correlation coefficient $r$ depends heavily on the true correlation $\rho$. For pairs of SNPs with very high or very low expected LD, fewer individuals are needed to confidently estimate $r^2$. The algorithm could adaptively evaluate a subset of $N' < N$ individuals, calculate a low-precision $r^2$, and only compute the full $N$-sample correlation if the intermediate value falls within a high-variance uncertainty threshold.
* **Lower Precision Arithmetic**: LD score regression relies on the aggregated sum of squares, making it exceptionally robust to minor numerical noise in individual $r^2$ estimates. Replacing `f64` with `f32`, or utilizing hardware-accelerated `bfloat16` for intermediate dot products, effectively doubles or quadruples SIMD lane utilization without materially affecting the statistical validity of the final heritability estimates.