Proliferation of non-linear excitations in the piecewise-linear  perceptron

Sclocchi, Antonio; Urbani, Pierfrancesco

doi:10.21468/scipostphys.10.1.013

Cited by 6 publications

(5 citation statements)

References 26 publications

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“…We therefore conjecture that the RSB phase is again isostatic and jamming critical. This would imply that we have another optimization problem for which the landscape is critical everywhere far from jamming as it happens for other cases [17][18][19].…”

Section: B the Distribution Of Gapsmentioning

confidence: 99%

High dimensional optimization under non-convex excluded volume constraints

Sclocchi,

Urbani

2021

Preprint

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We consider high dimensional random optimization problems where the dynamical variables are subjected to non-convex excluded volume constraints. We focus on the case in which the cost function is a simple quadratic cost and the excluded volume constraints are modeled by a perceptron constraint satisfaction problem. We show that depending on the density of constraints, one can have different situations. If the number of constraints is small, one typically has a phase where the ground state of the cost function is unique and sits on the boundary of the island of configurations allowed by the constraints. In this case there is an hypostatic number of constraints that are marginally satisfied. If the number of constraints is increased one enters in a glassy phase where the cost function has many local minima sitting again on the boundary of the regions of allowed configurations. At the phase transition point the total number of constraints that are marginally satisfied becomes equal to the number of degrees of freedom in the problem and therefore we say that these minima are isostatic. We conjecture that increasing further the constraints the system stays isostatic up to the point where the volume of available phase space shrinks to zero. We derive our results using the replica method and we also analyze a dynamical algorithm, the Karush-Kuhn-Tucker algorithm, through dynamical mean field theory and we show how to recover the results of the replica approach in the replica symmetric phase.

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Section: B the Distribution Of Gapsmentioning

confidence: 99%

High dimensional optimization under non-convex excluded volume constraints

Sclocchi,

Urbani

2021

Preprint

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“…This means that their response to perturbations becomes anomalous and driven by an abundance of soft modes. This approach has been successful to compute the values of the critical exponents of the jamming transition in hard spheres glasses [14] and beyond [15][16][17]. The Gardner transition therefore may provide the missing ingredient to understand the emergence of soft excitations in low temperature glasses from a first principle perspective.…”

Section: Introductionmentioning

confidence: 99%

Marginal stability of soft anharmonic mean field spin glasses

Folena¹,

Urbani²

2022

J. Stat. Mech.

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We investigate the properties of the glass phase of a recently introduced spin glass model of soft spins subjected to an anharmonic quartic local potential, which serves as a model of low temperature molecular or soft glasses. We solve the model using mean field theory and show that, at low temperatures, it is described by full replica symmetry breaking. As a consequence, at zero temperature the glass phase is marginally stable. We show that in this case, marginal stability comes from a combination of both soft linear excitations—appearing in a gapless spectrum of the Hessian of linear excitations—and pseudogapped non-linear excitations—corresponding to nearly degenerate two level systems. Therefore, this model is a natural candidate to describe what happens in soft glasses, where quasi localized soft modes in the density of states appear together with non-linear modes triggering avalanches and conjectured to be essential to describe the universal low temperature anomalies of glasses.

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“…Correspondingly their response become anomalous and driven by an abundance of soft modes. This picture has been extremely powerful to compute the values of the critical exponents of the jamming transition [5] and beyond [6][7][8] . While for a certain class of systems, namely colloidal glasses and relatives, signatures of the Gardner transition have been found in three dimensions [9][10][11][12][13][14][15][16][17] the Gardner transition in molecular glasses is more elusive [18,19].…”

Section: Introduction -mentioning

confidence: 99%

Marginal stability of soft anharmonic mean field spin glasses

Folena,

Urbani

2021

Preprint

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We investigate the properties of local minima of a recently introduced spin glass model of soft spins subjected to an anharmonic quartic local potential which serves as a model of low temperature molecular or soft glasses. We track the long time gradient descent dynamics in the glassy phase through dynamical mean field theory and show that spins are separated in two groups depending on their local stiffness. For spins having local stiffness that is right above its smallest possible value, the local fields distribution displays a depletion around the origin while those having a stiffness right below its largest possible value have a regular local fields distribution. We rationalize these findings through the replica method and show that the finite temperature phase transition to the glass phase is of continuous (full) replica-symmetry-breaking (RSB) type at all temperatures, down to zero temperature. Furthermore, marginal stability of the zero temperature fullRSB solution implies a linear pseudogap in the density of cavity fields for the spins with a local effective stiffness that is below a certain threshold. This generates a hole around the origin in the corresponding local field distribution. Those spins are natural candidates to model two level systems (TLS). The behavior of the cavity fields distribution for spins having stiffness close to the threshold one determines the tail of the low frequency density of states which is gapless. Therefore the spins having a stiffness around the cutoff one are the natural candidates to model quasi localized modes (QLM) in glasses.

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Proliferation of non-linear excitations in the piecewise-linear perceptron

Cited by 6 publications

References 26 publications

High dimensional optimization under non-convex excluded volume constraints

High dimensional optimization under non-convex excluded volume constraints

Marginal stability of soft anharmonic mean field spin glasses

Marginal stability of soft anharmonic mean field spin glasses

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