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 \end{align}\]</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/mlcolab/Pathfinder.jl/blob/8f220d225524429fbd082794bedfd5bd9d597107/src/inverse_hessian.jl#L68-L98">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Pathfinder.lbfgs_inverse_hessians-Tuple{Any, Any}" href="#Pathfinder.lbfgs_inverse_hessians-Tuple{Any, Any}"><code>Pathfinder.lbfgs_inverse_hessians</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">lbfgs_inverse_hessians(
     θs, ∇logpθs; Hinit=gilbert_init, history_length=5, ϵ=1e-12
 ) -&gt; Tuple{Vector{WoodburyPDMat},Int}</code></pre><p>From an L-BFGS trajectory and gradients, compute the inverse Hessian approximations at each point.</p><p>Given positions <code>θs</code> with gradients <code>∇logpθs</code>, construct LBFGS inverse Hessian approximations with the provided <code>history_length</code>.</p><p>The 2nd returned value is the number of BFGS updates to the inverse Hessian matrices that were rejected due to keeping the inverse Hessian positive definite.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/mlcolab/Pathfinder.jl/blob/8f220d225524429fbd082794bedfd5bd9d597107/src/inverse_hessian.jl#L12-L24">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Pathfinder.pdfactorize-Tuple{AbstractMatrix, AbstractMatrix, AbstractMatrix}" href="#Pathfinder.pdfactorize-Tuple{AbstractMatrix, AbstractMatrix, AbstractMatrix}"><code>Pathfinder.pdfactorize</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">pdfactorize(A, B, D) -&gt; WoodburyPDFactorization</code></pre><p>Factorize the positive definite matrix <span>$W = A + B D B^\mathrm{T}$</span>.</p><p>The result is the &quot;square root&quot; factorization <code>(L, R)</code>, where <span>$W = L R$</span> and <span>$L = R^\mathrm{T}$</span>.</p><p>Let <span>$U^\mathrm{T} U = A$</span> be the Cholesky decomposition of <span>$A$</span>, and let <span>$Q X = U^{-\mathrm{T}} B$</span> be a thin QR decomposition. Define <span>$C = I + XDX^\mathrm{T}$</span>, with the Cholesky decomposition <span>$V^\mathrm{T} V = C$</span>. Then, <span>$W = R^\mathrm{T} R$</span>, where</p><p class="math-container">\[R = \begin{pmatrix} U &amp; 0 \\ 0 &amp; I \end{pmatrix} Q^\mathrm{T} V.\]</p><p>The positive definite requirement is equivalent to the requirement that both <span>$A$</span> and <span>$C$</span> are positive definite.</p><p>For a derivation of this decomposition for the special case of diagonal <span>$A$</span>, see appendix A of <sup class="footnote-reference"><a id="citeref-Zhang2021" href="#footnote-Zhang2021">[Zhang2021]</a></sup>.</p><p>See <a href="#Pathfinder.pdunfactorize-Tuple{Pathfinder.WoodburyPDFactorization}"><code>pdunfactorize</code></a>, <a href="#Pathfinder.WoodburyPDFactorization"><code>WoodburyPDFactorization</code></a>, <a href="#Pathfinder.WoodburyPDMat"><code>WoodburyPDMat</code></a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/mlcolab/Pathfinder.jl/blob/8f220d225524429fbd082794bedfd5bd9d597107/src/woodbury.jl#L173-L200">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Pathfinder.pdunfactorize-Tuple{Pathfinder.WoodburyPDFactorization}" href="#Pathfinder.pdunfactorize-Tuple{Pathfinder.WoodburyPDFactorization}"><code>Pathfinder.pdunfactorize</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">pdunfactorize(F::WoodburyPDFactorization) -&gt; (A, B, D)</code></pre><p>Perform a reverse operation to <a href="#Pathfinder.pdfactorize-Tuple{AbstractMatrix, AbstractMatrix, AbstractMatrix}"><code>pdfactorize</code></a>.</p><p>Note that this function does not compute the inverse of <code>pdfactorize</code>. It only computes matrices that produce the same matrix <span>$W = A + B D B^\mathrm{T}$</span> as for the inputs to <code>pdfactorize</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/mlcolab/Pathfinder.jl/blob/8f220d225524429fbd082794bedfd5bd9d597107/src/woodbury.jl#L209-L217">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Pathfinder.resample-NTuple{4, Any}" href="#Pathfinder.resample-NTuple{4, Any}"><code>Pathfinder.resample</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">resample(rng, x, log_weights, ndraws) -&gt; (draws, psis_result)
-resample(rng, x, ndraws) -&gt; draws</code></pre><p>Draw <code>ndraws</code> samples from <code>x</code>, with replacement.</p><p>If <code>log_weights</code> is provided, perform Pareto smoothed importance resampling.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/mlcolab/Pathfinder.jl/blob/8f220d225524429fbd082794bedfd5bd9d597107/src/resample.jl#L1-L8">source</a></section></article><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-Byrd1994"><a class="tag is-link" href="#citeref-Byrd1994">Byrd1994</a>Byrd, R.H., Nocedal, J. &amp; Schnabel, R.B.          Representations of quasi-Newton matrices and their use in limited memory methods.          Mathematical Programming 63, 129–156 (1994).          doi: <a href="https://doi.org/10.1007/BF01582063">10.1007/BF01582063</a></li><li class="footnote" id="footnote-Zhang2021"><a class="tag is-link" href="#citeref-Zhang2021">Zhang2021</a>Lu Zhang, Bob Carpenter, Andrew Gelman, Aki Vehtari (2021).             Pathfinder: Parallel quasi-Newton variational inference.             arXiv: <a href="https://arxiv.org/abs/2108.03782">2108.03782</a> [stat.ML]</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../public/">« Public</a><a class="docs-footer-nextpage" href="../../examples/quickstart/">Quickstart »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Tuesday 3 October 2023 00:27">Tuesday 3 October 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+resample(rng, x, ndraws) -&gt; draws</code></pre><p>Draw <code>ndraws</code> samples from <code>x</code>, with replacement.</p><p>If <code>log_weights</code> is provided, perform Pareto smoothed importance resampling.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/mlcolab/Pathfinder.jl/blob/8f220d225524429fbd082794bedfd5bd9d597107/src/resample.jl#L1-L8">source</a></section></article><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-Byrd1994"><a class="tag is-link" href="#citeref-Byrd1994">Byrd1994</a>Byrd, R.H., Nocedal, J. &amp; Schnabel, R.B.          Representations of quasi-Newton matrices and their use in limited memory methods.          Mathematical Programming 63, 129–156 (1994).          doi: <a href="https://doi.org/10.1007/BF01582063">10.1007/BF01582063</a></li><li class="footnote" id="footnote-Zhang2021"><a class="tag is-link" href="#citeref-Zhang2021">Zhang2021</a>Lu Zhang, Bob Carpenter, Andrew Gelman, Aki Vehtari (2021).             Pathfinder: Parallel quasi-Newton variational inference.             arXiv: <a href="https://arxiv.org/abs/2108.03782">2108.03782</a> [stat.ML]</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../public/">« Public</a><a class="docs-footer-nextpage" href="../../examples/quickstart/">Quickstart »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Wednesday 4 October 2023 00:30">Wednesday 4 October 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>