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| -rw-r--r-- | marginal.tex | 60 |
1 files changed, 39 insertions, 21 deletions
diff --git a/marginal.tex b/marginal.tex index 099f603..fc89c5a 100644 --- a/marginal.tex +++ b/marginal.tex @@ -1275,7 +1275,8 @@ landscape: the problem of random nonlinear least squares optimization. Though, for reasons we will see it is easier to make predictions for random nonlinear \emph{most} squares, i.e., the problem of maximizing the sum of squared terms. We again take a spherical configuration space with $\mathbf x\in S^{N-1}$ and $0=g(\mathbf x)=\frac12(\|\mathbf x\|^2-N)$ as in the spherical spin glasses, and consider a set -of $M$ random functions $V_k:\mathbf S^{N-1}\to\mathbb R$ that are centered Gaussians with covariance +of $M=\alpha N$ random functions $V_k:\mathbf S^{N-1}\to\mathbb R$ that are +centered Gaussians with covariance \begin{equation} \overline{V_i(\mathbf x)V_j(\mathbf x')}=\delta_{ij}f\left(\frac{\mathbf x\cdot\mathbf x'}N\right) \end{equation} @@ -1344,12 +1345,14 @@ $\lambda^*$ is given by &=\int d\hat\beta\,d\hat\lambda\prod_{a=1}^n\lim_{m_a\to0}\prod_{\alpha=1}^{m_a}d\pmb\phi_a^\alpha \exp\left\{ \delta^{\alpha1}N(\hat\beta E+\hat\lambda\lambda^*) - -\frac12\int d1\,d2\,B^\alpha(1,2)\left[\sum_{k=1}^MV_k(\pmb\phi_a^\alpha)^2 - -\mu(\|\pmb\phi_a^\alpha\|^2-N)\right] + -\frac12\int d1\,d2\,\left[B^\alpha(1,2)\sum_{k=1}^MV_k(\pmb\phi_a^\alpha(1,2))^2 + -\mu\|\pmb\phi_a^\alpha(1,2)\|^2\right] \right\} \end{aligned} \end{equation} -The first step to evaluate this expression is to linearize the dependence on the random functions $V$. This is accomplished by inserting into the integral a Dirac $\delta$ function fixing the value of the energy for each replica, or +The first step to evaluate this expression is to linearize the dependence on +the random functions $V$. This is accomplished by inserting into the integral a +Dirac $\delta$ function fixing the value of the energy for each replica, or \begin{equation} \delta\big( V_k(\pmb\phi_a^\alpha(1,2))-v_{ka}^\alpha(1,2) @@ -1363,38 +1366,49 @@ The first step to evaluate this expression is to linearize the dependence on the where we have introduced auxiliary fields $\hat v$. With this inserted into the integral, all other instances of $V$ are replaced by $v$, and the only remaining dependence on the disorder is from the term $\hat vV$ arising from -the Fourier representation of the Dirac $\delta$ function. This term is linear in $V$, and therefore the random functions can be averaged over to produce +the Fourier representation of the Dirac $\delta$ function. This term is linear +in $V$, and therefore the random functions can be averaged over to produce \begin{equation} \overline{ \exp\left[ - i\sum_{ka\alpha}\int d1\,d2\,\hat v_{ka}^\alpha(1,2) + i\sum_k^M\sum_a^n\sum_\alpha^{m_a}\int d1\,d2\,\hat v_{ka}^\alpha(1,2) V_k(\pmb\phi_a^\alpha(1,2)) \right] } = - -\frac N2\sum_{ab}^n\sum_{\alpha\gamma}^{m_a}\sum_k^{\alpha N}\int d1\,d2\,d3\,d4\, - \hat v_{ka}^\alpha(1,2)f\big(\pmb\phi_a^\alpha(1,2)^T\pmb\phi_b^\gamma(3,4)\big)\hat v_{kb}^\gamma(3,4) + -\frac N2\sum_{ab}^n\sum_{\alpha\gamma}^{m_a}\sum_k^M\int d1\,d2\,d3\,d4\, + \hat v_{ka}^\alpha(1,2)f\big(\pmb\phi_a^\alpha(1,2)\cdot\pmb\phi_b^\gamma(3,4)\big)\hat v_{kb}^\gamma(3,4) \end{equation} -The entire integrand is now quadratic in the $v$ and $\hat v$ with the kernel +\end{widetext} +The entire integrand is now factorized in the indices $k$ and quadratic in the +$v$ and $\hat v$ with the kernel \begin{equation} \begin{bmatrix} - B_a^\alpha(1,2)\delta(1,3)\delta(2,4)\delta_{ab}\delta^{\alpha\gamma} & i\delta(1,3)\,\delta(2,4) \delta_{ab}\delta^{\alpha\gamma}\\ - i\delta(1,3)\,\delta(2,4) \delta_{ab}\delta^{\alpha\gamma}& f\big(\pmb\phi_a^\alpha(1,2)^T\pmb\phi_b^\gamma(3,4)\big) + B^\alpha(1,2)\delta(1,3)\delta(2,4)\delta_{ab}\delta^{\alpha\gamma} + & i\delta(1,3)\,\delta(2,4) \delta_{ab}\delta^{\alpha\gamma}\\ + i\delta(1,3)\,\delta(2,4) \delta_{ab}\delta^{\alpha\gamma} + & f\big(\pmb\phi_a^\alpha(1,2)\cdot\pmb\phi_b^\gamma(3,4)\big) \end{bmatrix} \end{equation} The integration over the $v$ and $\hat v$ results in a term in the effective action of the form -\begin{equation} - -\frac M2\log\operatorname{sdet}\left( - \delta(1,3)\,\delta(2,4) \delta_{ab}\delta^{\alpha\gamma} - +B_a^\alpha(1,2)f\big(\pmb\phi_a^\alpha(1,2)^T\pmb\phi_b^\gamma(3,4)\big) - \right) +\begin{equation} \label{eq:sdet.1} + \begin{aligned} + &-\frac M2\log\operatorname{sdet}\bigg[ + \delta(1,3)\,\delta(2,4) \delta_{ab}\delta^{\alpha\gamma} \\ + &\hspace{7em}+B^\alpha(1,2)f\big(\pmb\phi_a^\alpha(1,2)\cdot\pmb\phi_b^\gamma(3,4)\big) + \bigg] + \end{aligned} \end{equation} -When expanded, this supermatrix is constructed of the scalar products of the -real and Grassmann vectors that make up $\pmb\phi$. The change of variables to -these order parameters again results in the Jacobian of \eqref{eq:coordinate.jacobian}, contributing +When expanded, the supermatrix +$\pmb\phi_a^\alpha(1,2)\cdot\pmb\phi_b^\gamma(3,4)$ is constructed of the +scalar products of the real and Grassmann vectors that make up $\pmb\phi$. The +change of variables to these order parameters again results in the Jacobian of +\eqref{eq:coordinate.jacobian}, contributing \begin{equation} \frac N2\log\det J(C,R,D,Q,X,\hat X)-\frac N2\log\det G^2 \end{equation} +to the effective action. + Up to this point, the expressions above are general and independent of a given ansatz. However, we expect that the order parameters $X$ and $\hat X$ are zero, since this case is isotropic. Applying this ansatz here avoids a dramatically @@ -1405,7 +1419,9 @@ that $Q^{\alpha\gamma}=Q_{aa}^{\alpha\gamma}$ independently of the index $a$, implying that correlations in the tangent space of typical stationary points are the same. -Given these simplifying forms of the ansatz, taking the superdeterminant yields +Given these simplifying forms of the ansatz, taking the superdeterminant in +\eqref{eq:sdet.1} yields +\begin{widetext} \begin{equation} \begin{aligned} \log\det\left\{ @@ -1418,6 +1434,7 @@ Given these simplifying forms of the ansatz, taking the superdeterminant yields -2\log\det(I+G\odot f'(C)) \end{aligned} \end{equation} +\end{widetext} where once again $\odot$ is the Hadamard product and $A^{\circ n}$ gives the Hadamard power of $A$. We can already see one substantive difference between the structure of this problem and that of the spherical models: the effective @@ -1438,6 +1455,7 @@ We further take a planted replica symmetric structure for the matrix $Q$, identical to that in \eqref{eq:Q.structure}. The resulting effective action is the same as if we had made an annealed calculation in the complexity, though the previous expressions are general. +\begin{widetext} \begin{equation} \begin{aligned} \mathcal S_\beta @@ -1861,7 +1879,7 @@ dominant optima in the random nonlinear least squares problem of section \ref{sec:least.squares}. While in this paper we only treat problems with a replica symmetric structure, formulas for the effective action are generic to any structure and provide a starting point for analyzing the challenging -full-RSB setting. +full \textsc{rsb} setting. Using the $\mathbb R^{N|2}$ superfields \begin{equation} |