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Before reading this section, ensure you are familiar with the Linear Algebra Annex. After reading this section, consider doing this assignment.

Assignment

Explore low-rank Gaussian geometry with guided exercises
Large linear operators appear throughout modern machine learning systems. Yet empirical evidence shows that meaningful changes to such systems often occupy only a small number of directions. This tutorial develops that idea from first principles, using Gaussian distributions to isolate the geometry of low-rank change before connecting it to modern architectures like LoRA (Low-Rank Adaptation).

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Part I: Covariance and Geometry

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Covariance as shape

Let x∈Rdx \in \mathbb{R}^d be a zero-mean random vector with covariance Σ=E[xx⊤].\Sigma = \mathbb{E}[x x^\top]. Eigenvectors of Σ\Sigma define orthogonal directions in space, and eigenvalues measure the variance along those directions. Geometrically, Σ\Sigma describes the shape of a probability ellipsoid.
Internalize that eigenvalues are not abstract quantities - they directly correspond to observable spread in data.

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Latent variable construction

Instead of specifying Σ\Sigma directly, we construct it from a lower-dimensional latent variable: z∼N(0,Ik),x=Wz+ε,ε∼N(0,σ2Id).z \sim \mathcal{N}(0, I_k), \quad x = W z + \varepsilon, \quad \varepsilon \sim \mathcal{N}(0, \sigma^2 I_d). This implies Σ=WW⊤+σ2Id.\Sigma = W W^\top + \sigma^2 I_d. The matrix WW determines the signal subspace; the noise term fills the remaining directions isotropically.
This is the same construction used in classical factor analysis and probabilistic PCA.

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Rank as a geometric constraint

Because rank(WW⊤)≤k\mathrm{rank}(W W^\top) \le k, only kk directions can carry variance above the noise level. This explains why many high-dimensional datasets exhibit:
  • Rapidly decaying eigenvalue spectra
  • Effective low dimensionality

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Part II: Structured Coefficient Changes

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A reference distribution

Fix a reference matrix W0W_0 with covariance Σ0=W0W0⊤+σ2Id.\Sigma_0 = W_0 W_0^\top + \sigma^2 I_d. This defines a baseline Gaussian distribution. All subsequent changes are measured relative to this geometry.

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Low-rank coefficient modifications

We now restrict changes to the form ΔW=BA,\Delta W = B A, where:
  • BB is a d×rd \times r matrix
  • AA is an r×kr \times k matrix
  • r≪min⁡(d,k)r \ll \min(d,k)
Only rr new directions can be introduced.
Low-rank structure does not limit how much the matrix changes, but where it can change.

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Diffuse coefficient changes

For comparison, consider dense changes with no preferred directions, scaled to match the Frobenius norm of the low-rank case. This contrast isolates the role of structure from magnitude.

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Observable consequences

Empirically, one finds:
  • Low-rank changes alter a small number of eigenvalues dramatically
  • Diffuse changes alter many eigenvalues modestly
This establishes the central principle:
Rank limits the number of variance directions that can be modified.

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Part III: Likelihood and Statistical Geometry

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Empirical covariance

Given samples {xi}i=1n\{x_i\}_{i=1}^n, the empirical covariance S=1n∑i=1nxixi⊤S = \frac{1}{n} \sum_{i=1}^n x_i x_i^\top is a sufficient statistic for zero-mean Gaussian models.

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Gaussian likelihood geometry

The Gaussian negative log-likelihood is L(Σ)=12[log⁡det⁡Σ+tr ⁣(Σ−1S)].\mathcal{L}(\Sigma) = \frac{1}{2} \left[ \log \det \Sigma + \mathrm{tr}\!\left(\Sigma^{-1} S\right) \right]. Geometric interpretation:
  • log⁡det⁡Σ\log \det \Sigma penalizes volume mismatch
  • tr(Σ−1S)\mathrm{tr}(\Sigma^{-1} S) penalizes directional mismatch
You may recognize this as a Riemannian geometry on the cone of positive definite matrices.

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Rank-constrained covariance matching

Expanding Σ−Σ0=W0ΔW⊤+ΔWW0⊤+ΔWΔW⊤\Sigma - \Sigma_0 = W_0 \Delta W^\top + \Delta W W_0^\top + \Delta W \Delta W^\top implies rank(Σ−Σ0)≤2r.\mathrm{rank}(\Sigma - \Sigma_0) \le 2r. Thus a rank-rr coefficient change can only modify a limited number of eigen-directions, regardless of dimensionality. This fact explains the likelihood saturation observed in experiments.

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Part IV: Connection to LoRA in Transformers

LoRA (Low-Rank Adaptation) applies the same structural assumption to large linear operators:
  • A reference matrix is fixed (the pretrained weights)
  • Adaptation is constrained to a low-rank subspace
  • Rank controls expressive capacity
Transformers obscure this geometry with nonlinearities and attention mechanisms. The Gaussian setting exposes it directly. 
# LoRA structure in practice
# Original: y = Wx
# LoRA:     y = Wx + BAx  where B ∈ R^{d×r}, A ∈ R^{r×k}

class LoRALayer:
    def __init__(self, W, rank):
        self.W = W  # frozen pretrained weights
        self.B = initialize_zeros(W.shape[0], rank)
        self.A = initialize_random(rank, W.shape[1])

    def forward(self, x):
        return self.W @ x + self.B @ (self.A @ x)

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Final Takeaway

Low-rank adaptation is not an optimization trick. It is a geometric assumption about how complex systems change.
  • When that assumption holds, low-rank methods are statistically efficient
  • When it does not, no algorithm can avoid higher-dimensional modification

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References

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