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| -rw-r--r-- | marginal.bib | 99 | ||||
| -rw-r--r-- | marginal.tex | 13 |
2 files changed, 53 insertions, 59 deletions
diff --git a/marginal.bib b/marginal.bib index 3e9eb8b..784d3f0 100644 --- a/marginal.bib +++ b/marginal.bib @@ -239,12 +239,10 @@ @unpublished{ElAlaoui_2020_Algorithmic, author = {El Alaoui, Ahmed and Montanari, Andrea}, title = {Algorithmic Thresholds in Mean Field Spin Glasses}, - year = {2020}, - month = {Sept}, url = {http://arxiv.org/abs/2009.11481v1}, archiveprefix = {arXiv}, date = {2020-09-24T04:22:42Z}, - eprint = {2009.11481v1}, + eprint = {2009.11481}, eprintclass = {cond-mat.stat-mech}, eprinttype = {arxiv}, primaryclass = {cond-mat.stat-mech} @@ -515,29 +513,23 @@ @unpublished{Huang_2023_Algorithmic, author = {Huang, Brice and Sellke, Mark}, title = {Algorithmic Threshold for Multi-Species Spherical Spin Glasses}, - year = {2023}, - month = {mar}, - url = {http://arxiv.org/abs/2303.12172v2}, + url = {http://arxiv.org/abs/2303.12172}, archiveprefix = {arXiv}, - date = {2023-03-21T20:09:08Z}, - eprint = {2303.12172v2}, + eprint = {2303.12172}, eprintclass = {math.PR}, - eprinttype = {arxiv}, - urldate = {2024-06-13T13:10:56.404805Z} + primaryclass = {math.PR}, + eprinttype = {arxiv} } @unpublished{Huang_2023_Strong, author = {Huang, Brice and Sellke, Mark}, title = {Strong Topological Trivialization of Multi-Species Spherical Spin Glasses}, - year = {2023}, - month = {aug}, - url = {http://arxiv.org/abs/2308.09677v2}, + url = {http://arxiv.org/abs/2308.09677}, archiveprefix = {arXiv}, - date = {2023-08-18T16:56:19Z}, - eprint = {2308.09677v2}, + eprint = {2308.09677}, eprintclass = {math.PR}, - eprinttype = {arxiv}, - urldate = {2024-06-13T13:07:13.561947Z} + primaryclass = {math.PR}, + eprinttype = {arxiv} } @article{Huang_2024_Optimization, @@ -600,13 +592,11 @@ @unpublished{Kamali_2023_Stochastic, author = {Kamali, Persia Jana and Urbani, Pierfrancesco}, title = {Stochastic Gradient Descent outperforms Gradient Descent in recovering a high-dimensional signal in a glassy energy landscape}, - year = {2023}, - month = {sep}, url = {http://arxiv.org/abs/2309.04788v2}, - note = {}, archiveprefix = {arXiv}, - eprint = {2309.04788v2}, + eprint = {2309.04788}, eprintclass = {cs.LG}, + primaryclass = {cs.LG}, eprinttype = {arxiv} } @@ -627,11 +617,10 @@ @unpublished{Kent-Dobias_2024_Algorithm-independent, author = {Kent-Dobias, Jaron}, title = {Algorithm-independent bounds on complex optimization through the statistics of marginal optima}, - year = {2024}, url = {https://arxiv.org/abs/2407.02092}, archiveprefix = {arXiv}, - eprint = {2407.02092}, - primaryclass = {cond-mat.dis-nn} + primaryclass = {cond-mat.dis-nn}, + eprint = {2407.02092} } @article{Kent-Dobias_2024_Arrangement, @@ -652,7 +641,6 @@ @unpublished{Kent-Dobias_2024_Conditioning, author = {Kent-Dobias, Jaron}, title = {Conditioning the complexity of random landscapes on marginal optima}, - year = {2024}, url = {https://arxiv.org/abs/2407.02082}, archiveprefix = {arXiv}, eprint = {2407.02082}, @@ -765,26 +753,23 @@ @unpublished{Montanari_2023_Solving, author = {Montanari, Andrea and Subag, Eliran}, title = {Solving overparametrized systems of random equations: I. Model and algorithms for approximate solutions}, - year = {2023}, - month = {jun}, url = {http://arxiv.org/abs/2306.13326v1}, note = {}, archiveprefix = {arXiv}, - eprint = {2306.13326v1}, + eprint = {2306.13326}, eprintclass = {math.PR}, + primaryclass = {math.PR}, eprinttype = {arxiv} } @unpublished{Montanari_2024_On, author = {Montanari, Andrea and Subag, Eliran}, title = {On {Smale}'s 17th problem over the reals}, - year = {2024}, - month = {may}, - url = {http://arxiv.org/abs/2405.01735v1}, - note = {}, + url = {http://arxiv.org/abs/2405.01735}, archiveprefix = {arXiv}, - eprint = {2405.01735v1}, + eprint = {2405.01735}, eprintclass = {cs.DS}, + primaryclass = {cs.DS}, eprinttype = {arxiv} } @@ -835,11 +820,9 @@ @unpublished{Parisi_1995-01_On, author = {Parisi, Giorgio}, title = {On the Statistical Properties of the Large Time Zero Temperature Dynamics of the {SK} Model}, - year = {1995}, - month = {jan}, - url = {http://arxiv.org/abs/cond-mat/9501045v1}, + url = {http://arxiv.org/abs/cond-mat/9501045}, archiveprefix = {arXiv}, - eprint = {cond-mat/9501045v1}, + eprint = {cond-mat/9501045}, eprinttype = {arxiv} } @@ -887,12 +870,11 @@ @unpublished{Shklovskii_2024_Half, author = {Shklovskii, B. I.}, title = {Half century of {Efros}-{Shklovskii} {Coulomb} gap. Romance with {Coulomb} interaction and disorder}, - year = {2024}, - month = {mar}, - url = {http://arxiv.org/abs/2403.19793v5}, + url = {http://arxiv.org/abs/2403.19793}, archiveprefix = {arXiv}, - eprint = {2403.19793v5}, + eprint = {2403.19793}, eprintclass = {cond-mat.mtrl-sci}, + primaryclass = {cond-mat.mtrl-sci}, eprinttype = {arxiv} } @@ -928,15 +910,12 @@ @unpublished{Subag_2021_TAP, author = {Subag, Eliran}, title = {{TAP} approach for multi-species spherical spin glasses {I}: general theory}, - year = {2021}, - month = {nov}, url = {http://arxiv.org/abs/2111.07132v1}, archiveprefix = {arXiv}, - date = {2021-11-13T15:21:40Z}, - eprint = {2111.07132v1}, + eprint = {2111.07132}, eprintclass = {math.PR}, - eprinttype = {arxiv}, - urldate = {2024-06-13T13:04:28.790463Z} + primaryclass = {math.PR}, + eprinttype = {arxiv} } @article{Subag_2023_TAP, @@ -992,25 +971,22 @@ @unpublished{Urbani_2024_Statistical, author = {Urbani, Pierfrancesco}, title = {Statistical physics of complex systems: glasses, spin glasses, continuous constraint satisfaction problems, high-dimensional inference and neural networks}, - year = {2024}, - month = {may}, - url = {http://arxiv.org/abs/2405.06384v1}, - note = {}, + url = {http://arxiv.org/abs/2405.06384}, archiveprefix = {arXiv}, - eprint = {2405.06384v1}, + eprint = {2405.06384}, eprintclass = {cond-mat.dis-nn}, + primaryclass = {cond-mat.dis-nn}, eprinttype = {arxiv} } @unpublished{Vivo_2024_Random, author = {Vivo, Pierpaolo}, title = {Random Linear Systems with Quadratic Constraints: from Random Matrix Theory to replicas and back}, - year = {2024}, - month = {jan}, - url = {http://arxiv.org/abs/2401.03209v2}, + url = {http://arxiv.org/abs/2401.03209}, archiveprefix = {arXiv}, - eprint = {2401.03209v2}, + eprint = {2401.03209}, eprintclass = {cond-mat.stat-mech}, + primaryclass = {cond-mat.stat-mech}, eprinttype = {arxiv} } @@ -1082,3 +1058,14 @@ issn = {1742-5468} } +@unpublished{Kent-Dobias_2024_On, + author = {Kent-Dobias, Jaron}, + title = {On the topology of solutions to random continuous constraint satisfaction problems}, + url = {http://arxiv.org/abs/2409.12781}, + archiveprefix = {arXiv}, + eprint = {2409.12781}, + eprintclass = {cond-mat.dis-nn}, + primaryclass = {cond-mat.dis-nn}, + eprinttype = {arxiv} +} + diff --git a/marginal.tex b/marginal.tex index d9e1d47..b9fabc9 100644 --- a/marginal.tex +++ b/marginal.tex @@ -469,8 +469,9 @@ extremizing the Lagrangian L(\mathbf x,\pmb\omega)=H(\mathbf x)+\sum_{i=1}^r\omega_ig_i(\mathbf x) \end{equation} with respect to $\mathbf x$ and the Lagrange multipliers -$\pmb\omega=\{\omega_1,\ldots,\omega_r\}$. The corresponding gradient and -Hessian of the energy associated with this constrained extremal problem are +$\pmb\omega=\{\omega_1,\ldots,\omega_r\}$. To write the gradient and Hessian of the energy, which are necessary to count stationary points, care must be taken to ensure they are constrained to the tangent space of the configuration manifold. For our purposes, the Lagrangian formalism offers a solution: the gradient $\nabla H:\mathbb R^N\times\mathbb R^r\to\mathbb R^N$ and +Hessian $\operatorname{Hess} H:\mathbb R^N\times\mathbb R^r\to\mathbb R^{N\times N}$ of the energy $H$ can be written as the simple vector derivatives of +the Lagrangian $L$, with \begin{align} &\nabla H(\mathbf x,\pmb\omega) =\partial L(\mathbf x,\pmb\omega) @@ -483,7 +484,13 @@ Hessian of the energy associated with this constrained extremal problem are \end{aligned} \end{align} where $\partial=\frac\partial{\partial\mathbf x}$ will always represent the -derivative with respect to the vector argument $\mathbf x$. +derivative with respect to the vector argument $\mathbf x$. Note that unlike +the energy, which is a function of the configuration $\mathbf x$ alone, the +gradient and Hessian depend also on the Lagrange multipliers $\pmb\omega$. In situations +with an extensive number of constraints, it is important to take seriously +contributions of the form $\frac{\partial^2L}{\partial\mathbf +x\partial\pmb\omega}$ to the Hessian \cite{Kent-Dobias_2024_On}. However, the cases we study here have +$N^0$ constraints and these contributions appear as finite-$N$ corrections. The number of stationary points in a landscape for a particular function $H$ is found by |