Add some statistics formulas to problem.js

[ddavidd23]

I added the following three statistics-related formulas which have interesting LaTeX formulas:

1. Ridge regression: (\mathcal{L}_X = \mathrm{d} \circ \iota_X + \iota_X \circ \mathrm{d})
2. ELBO: (L(\phi,\theta;x) = \mathbb{E}{z \sim q\phi\left(\cdot\mid x\right)} \left[ \ln\frac{p_\theta(x,z)}{q_\phi(z\mid x)} \right])
3. Langevin dynamics: (\mathrm{d}x_t = -\nabla U(x_t) \mathrm{d}t + \sqrt{2\beta^{-1}} \mathrm{d}W_t)

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Important

Add Ridge Regression, ELBO, and Langevin Dynamics formulas to problems.js.

• Additions to problems.js:
  • Add "Ridge Regression" with L2 regularization formula.
  • Add "Evidence Lower Bound (ELBO)" for log-likelihood of observed data.
  • Add "Langevin Dynamics (Overdamped)" for stochastic gradient flow.

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[ellipsis-dev]

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1. public/assets/js/problems.js:915

• Draft comment:
  The PR description lists the Ridge Regression formula as Cartan's magic formula, but the code uses a different (and correct) expression. Please update the PR description to match the code.
• Reason this comment was not posted:
  Comment did not seem useful. Confidence is useful = 0% <= threshold 50% The comment is asking the PR author to update the PR description, which is against the rules. The focus should be on the code itself, not the PR description.

2. public/assets/js/problems.js:916

• Draft comment:
  Consider clarifying the notation in the Ridge Regression formula (e.g. using explicit multiplication '\�5{X}\mathbf{w}' if desired) for consistency.
• Reason this comment was not posted:
  Confidence changes required: 33% <= threshold 50% None

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