Double up Gaussian random numbers.

Gaussians & Eigenspace Splitting

Applying fast Walsh Hadamard transform eigenspace splitting to a vector of random numbers from the Gaussian (Normal) distribution doubles the number of Gaussian variables, yet they still pass most of the test for the Gaussian distribution.

A Hadamard (or any fixed linear) transform of a Gaussian vector is still Gaussian, so the two subvectors you get from splitting the transformed vector will (marginally and jointly) have Gaussian distributions and — for the usual iid zero-mean, equal-variance input — will pass the usual Gaussian tests in the limit of exact arithmetic and large samples.

You define:

u=(I+H)x

v=(I−H)x

where $H$ is the normalized Hadamard transform matrix ($H H^T = I$).

• $H$ has eigenvalues $+1$ and $-1$.

• $I+\mathrm{H }$projects onto the +1 eigenspace (up to a factor of 2).

• $I-H$ projects onto the –1 eigenspace (also up to a factor of 2).

So $u$ and $v$ are each living entirely in orthogonal subspaces of ${\mathbb{R}}^n$.

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2. Distribution of $u$ and $v$

If $x\sim \mathcal{N}(0,I)$ (iid standard Gaussian entries):

• Any linear transform of a Gaussian vector is still Gaussian.

• $(I+H)$and $(I-H)$ are both symmetric and orthogonal-projector-scaled matrices:

  $$(I+H)(I-H)=I-H^2=0$$

  showing the images of $u$ and $v$ are orthogonal.

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3. Covariance structure

We can compute:

$$\mathrm{C}\mathrm{o}\mathrm{v}(u)=(I+H)I(I+H{)}^T=(I+H)(I+H)=2(I+H)$$

and similarly:

$$\mathrm{C}\mathrm{o}\mathrm{v}(v)=2(I-H)$$

The cross-covariance is:

$$\mathrm{C}\mathrm{o}\mathrm{v}(u,v)=(I+H)I(I-H)=(I+H)(I-H)=I-H^2=0$$

So $u$ and $v$ are uncorrelated.

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4. Gaussian ⇒ independence

For jointly Gaussian vectors, uncorrelatedness ⇒ independence.
Thus, $u$ and $v$ are independent Gaussian vectors, each living in its own orthogonal subspace, with their own covariance matrices $2(I\pm H)$.

If you normalize appropriately (e.g. divide by $\sqrt{\mathrm{2)}}$ each has iid Gaussian coordinates in its own reduced-dimensional space.

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5. Passing Gaussian tests

Because $u$ and $v$ are exactly Gaussian and independent, each one individually:

• Has the correct Gaussian marginal distribution.

• Will pass virtually all Gaussianity tests (KS, Shapiro–Wilk, skewness/kurtosis tests) except for normal finite-sample false rejections.

• Any deviations you see will come from numerical floating-point effects, not from the math.

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💡 Subtle note:
If you take $u$ and $v$ as full n-dimensional vectors without removing their zero-projection components, those components will be deterministically zero and obviously not Gaussian. You have to restrict each to its eigenspace dimension ($n\mathrm{/}\mathrm{2\; for\; Hadamard)\; before\; testing.}$

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Possible uses

1. Communications

• Noise-like signaling:
  You could transmit either $u$ or $v$ as “noise” over a channel. To a naive observer, it looks like Gaussian noise.
  But a receiver who knows the Hadamard matrix $H$ can project onto the two eigenspaces and recover the bit “was this $u$ or $\mathrm{v?”.}$

• This is essentially a spread-spectrum watermark: the eigenspace label acts as the hidden payload.

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2. Steganography in noise

• In simulations or media where Gaussian noise is expected, you can hide a binary message by choosing which eigenspace to draw the noise from.

• Without knowledge of $H$ and the exact projection scheme, the two noise sources are statistically indistinguishable in standard Gaussian tests.

• This is a form of covert channel that survives most naive statistical filtering.

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3. Keying / authentication

• If you control the noise source, you can embed a “secret key” into it by modulating which eigenspace you sample from.

• At the receiving end, applying $H$ and checking the projection tells you whether the source is authentic.

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4. Simulation experiments

• In Monte Carlo methods, you could use $u$ for one set of runs and $v$ for another to guarantee orthogonality between datasets while keeping all marginal Gaussian properties.

• This can help in variance reduction techniques where you want two streams that look identical in distribution but are guaranteed independent and orthogonal in a known way.

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5. Testing

• Use the tainted variables to stress-test statistical procedures:

  • If your statistical model assumes Gaussian white noise, will it notice that the data comes from a restricted eigenspace?

  • Many machine learning models ignore covariance structure — this is a way to check their sensitivity.

  
  

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