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Riccardo Finotello 056c983967 Shorter presentation and change in the last slide 
Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
2020-12-17 19:30:27 +01:00

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- page 1/102
Today I'll talk about my work as a Ph.D. student at the University of Torino.
I will deal with aspects related to phenomenology from the point of view of
strings for which I worked both on theoretical and computational levels.
- page 2/102
The plan is to first introduce a semi-phenomenological description of physics
from a theory of strings.
These tools will be used throughout the talk as basis for the rest of the
topics.
I will then mainly deal with open strings and analyse correlators in the
presence of twist and spin fields seen as singular points in the string
propagation.
From particle physics, I will then move to a string theory description of
cosmology in the presence of time dependent singularities in time dependent
orbifods.
Finally I will focus on computational aspects of string compactifications.
- page 3/102
I will therefore start with the most geometrical aspects of the discussion,
reviewing some basics and introducing a framework to deal with open string
amplitudes in the presence of twist and spin fields.
- page 4/102
As usual in string theory the starting point is Polyakov's action.
We start directly from the superstring extension where gamma is the
worldsheet metric (for the moment generic) and rhos are the 2-dimensional
"gamma matrices".
- page 5/102
This description presents lots of symmetries which, at the end of the day,
provide the key to its success.
In fact the invariances of the action are such that the theory is conformal
with a vanishing traceless stress tensor.
- page 6/102
Using standard field theory methods the stress energy tensor of the
superstring in D dimensions generates the known Virasoro algebra, extending
the classical de Witt's algebra.
At quantum level it presents a central charge whose value depends on the
dimension of the target space.
- page 9/102
In order for the stress energy tensor to be a conformal primary field, we
need to introduce sets of fields known as conformal ghosts.
These are conformal fields specified by their weight lambda and they are
introduced as a first order Lagrangian theory.
- page 8/102
The central charge arising from the algebra of the modes of the ghost sector
can then be used to compensate the superstring counterpart.
Consistency of the theory thus fixes the dimension of the target space to be
a 10-dimensional spacetime.
- page 9/102
The fact that "everyday" physics in accelerators is 4-dimensional is
recovered through compactification.
In simple words we recover 4-dimensional physics at low energy by considering
the 10-dimensional Minkowski space as a product of the usual 4-dimensional
spacetime and a 6-(real)-dimensional space.
The internal space has to obey stringent restrictions: in particular
it has to be a compact manifold (to "hide" the extra dimensions), it has to
break most of the supersymmetry present at high energy, and its arising gauge
algebra has to contain the standard model algebra.
- page 10/102
These manifolds have been firstly postulated by Eugenio Calabi and later
proved to exist by Shing Tung Yau, hence the name Calabi-Yau manifolds.
In the case at hand, they are a specific class of 3-dimensional complex
manifolds with SU(3) holonomy.
They must be Ricci-flat or equivalently have a vanishing first Chern class.
- page 11/102
In general it is not easy to classify these manifolds (as well as computing
for instance their metric, which is generally not known).
However, for instance, we will see that the dimension of the complex
cohomology groups, known as Hodge numbers, will play a strategic role.
- page 12/102
We solve the equations of motion and the boundary conditions for the string
propagation (we focus on the bosonic part for the moment).
The action in fact introduces naturally the Neumann boundary conditions for
the strings.
The solution factorises into its holomorphic components (where z is the usual
coordinate on the complex plane) due to the equations of motion.
- page 13/102
Now consider for a second the simplest toroidal compactification of closed
strings, that is suppose we can "hide" the last direction of the string on a
circle.
As in usual quantum mechanics this leads to momentum quantization (defined as
an integer n in units of the compactification radius).
A closed string can moreover wind an integer number of times around the cycle
introducing a "winding number" m
In turn this is reflected on the spectrum of the theory.
Differently from field theory, shrinking the radius does not decouple the
modes, but the compact dimension remains.
In other words exchanging R and 1/R does not modify the theory.
- page 14/102
This so called T-duality can also be applied to open strings with a different
outcome.
Since open strings cannot wind around the compact dimension, the behaviour of
the spectrum is as in field theory: the compact dimension decouples and the
open string is constrained on a lower dimensional surface.
This is the result of introducing Dirichlet boundary conditions on the T-dual
coordinate, meaning that the endpoints of the string have to reside on the
same surface.
The procedure can be applied to several dimensions thus introducing surfaces
on which the endpoints of the open string live, called D-branes.
- page 15/102
D-branes naturally introduce preferred directions of motion thus breaking the
D-dimensional Poincaré symmetry.
It is in fact possible to show that in D-dimensions, the open string sector
at massless level contains an Abelian field.
D-branes split the components into a lower dimensional U(1) field on the
D-brane and a vector of spacetime scalars.
- page 16/102
In general we can have strings whose endpoints are on the same D-brane
leading to U(1) gauge theories as well as stretched strings across different
branes.
The Chan-Paton factors can be used to label the endpoints of the strings by
the position on the D-branes.
It is then possible to show that when the D-branes are coincident, states can
be rearranged into enhanced gauge groups (for instance, unitary), thus
creating non Abelian gauge theories.
- page 17/102
However, physical constraints on the possible constructions pose serious
questions on the D-brane dispositions.
A quark can be modelled with a string stretched across two distant branes,
but such quark would have a mass proportional to the distance to the branes:
chirality in particular would not be possible to define, while being one of
the defining features of the Standard Model.
This is where the possibility to put D-branes at an angle with respect to
each other becomes crucial.
While most of the modes will indeed gain a mass, the massless spectrum can
support chiral states localised at the intersections.
- page 18/102
In this framework we consider models built with D6-branes.
We embed them in 6-dimensional internal space without worrying about
compactification for now.
In this scenario we study correlators of fields in the presence of twist
fields, which are a set of conformal fields arising at the intersections.
These are particularly interesting to compute for instance Yukawa couplings
in string theory.
- page 19/102
Using the path integral approach, correlators involving twist fields are
dominated by the instanton contribution of the classical Euclidean action.
We focus specifically on its computation in the case of three D-branes
necessary for the Yukawa couplings.
- page 20/102
The literature already takes into consideration D-branes embedded as lines in
a factorised version of the internal space, where the possible relative
rotations are Abelian in each plane of the space.
The contribution of the string in this case is proportional to the area of
the triangle formed on the plane by the three D-branes since the
string is completely constrained to move on the plane.
- page 21/102
In what follows we consider a non completely factorised space and we focus on
its 4-dimensional sector.
After filling the physical 4-dimensional space, we study the remaining
directions of the D-brane in 4-dimensional internal space using a
well-adapted frame of reference which is in general rotated with respect to
global coordinates.
The rotation is not directly an SO(4) element, but it is an element of a
Grassmannian: in fact separately rotating the Dirichlet and Neumann
components does not change anything (we just need to relabel the
coordinates), as long as no reflections are involved.
The rotation involved is therefore a representative of a left equivalence
class of possible rotations.
- page 22/102
Introducing the usual conformal transformations mapping the strip to the
complex plane, the intersecting branes are mapped to the real axis.
D-branes are therefore real intervals on the space, between their
intersection points.
- page 23/102
In this definition, the boundary conditions of the strings become a set of
discontinuities on the real axis, one for each D-brane, and the embedding
equation specifying the intersections.
- page 24/102
Instead of dealing directly with the discontinuities, it is more suitable to
introduce an auxiliary function defined over the entire complex plane by
gluing functions defined on the upper plane and on the bottom plane on
arbitrary interval.
This "doubling trick" transforms the discontinuity into a monodromy factor,
when looping any base point through the glued interval.
- page 25/102
The fact that rotations are non Abelian leads to two different monodromies
for a base points starting in the upper plane or in the bottom plane.
This is a general feature, but the 3 D-branes case simplifies this enough.
- page 26/102
Dealing with these 4 x 4 matrices is delicate.
Using a known isomorphism between SO(4) matrices and two copies of SU(2) x
SU(2), the monodromy matrix can be cast into a tensor product of two 2 x 2
matrices.
In this matrix form the solution is therefore given by 2-dimensional basis of
holomorphic functions with three regular singular points.
- page 27/102
The usual SL invariance allows us to fix the monodromies in 0, 1 and
infinity.
The overall solution will then be a superposition of all possible basis.
Given the previous properties we therefore look for a basis of Gauss
hypergeometric functions.
- page 28/102
Since we deal with rotations, the parameters of the hypergeometric functions
involved are indeed connected to the rotation vectors.
However the choice is not unique and labelled by the
periodicity of the rotations.
- page 29/102
The reason is a huge redundancy in the description: using the free parameters
of the rotations we should in fact fix all degrees of freedom in the
solution, which at the moment is an infinite sum involving an infinite amount
of free parameters.
In fact we only showed that the rotation matrix is equivalent to a
monodromy matrix from which can build an overparametrised solution.
Using contiguity relations we can then restrict the sum over independent
functions.
Finally requiring the Euclidean action to be finite restricts the sum to only
two terms (the particular terms surviving in the sum depend on the rotation
vectors but they are never more than two).
Fixing the intersection points eventually determines the free constants in the solution.
- page 30/102
The physical interpretation of the solution is finally straightforward in the
Abelian case, where the action can be reduced to the sum of the areas of the
internal triangles (this is a general result even for a generic number of
D-branes).
- page 31/102
In the non Abelian case we considered there is no simple way to write the
action using global data.
However the contribution to the Euclidean action is larger than the Abelian
case: the strings are in fact no longer constrained on a plane and, in order
to stretch across the boundaries, they have to form a small bump while
detaching from the D-brane.
The Yukawa coupling in this case is therefore suppressed with respect to the
Abelian case.
- page 32/102
We then turn the attention to fermions and the computation of correlators
involving spin fields.
Though ideally extending some ideas, we abandon the intersecting D-brane
scenario, and we introduce point-like defects on one boundary of the
worldsheet in such a way that the superstring undergoes a change of its
boundary conditions when meeting a defect.
- page 33/102
It is possible to show that in this case the Hamiltonian of the theory
is only piecewise conserved.
- page 34/102
Suppose now that we could expand the field on a basis of solutions to the
boundary conditions and work, as before, on the entire complex plane.
- page 35/102
Ideally we would be interested in extracting the modes in order to perform
any computation of amplitudes.
The definition of the operation is connected to a dual basis whose form is
completely fixed by the original field (which we know) and the request of
time independence.
- page 36/102
The resulting algebra of the operators is in fact defined through such
operation and it is therefore time independent.
- page 37/102
Differently from what done in the bosonic case, we focus on U(1) boundary
change operators.
The resulting monodromy on the complex plane is therefore a phase factor.
- page 38/102
As in the previous case we can write a basis of solutions which incorporates
the behaviour looping the point-like defects.
Consequently we can also define a dual basis.
Both fields are defined up to integer factors, since we are still dealing
with rotations.
- page 39/102
In order to compute amplitudes we then need to define the space on which the
representation of the algebra acts.
We define an excited vacuum, annihilated by positive frequency modes, and the
lowest energy vacuum (from the strip definition).
- page 40/102
Vacua need to be consistent, leading to conditions labelled by an integer
factor relating the basis of solutions with its dual (and ultimately the
algebra of operators).
The vacuum must always be correctly normalised and the description of physics
using any two of the vacuum definitions should be consistently equivalent.
- page 41/102
To avoid having overlapping in- and out-annihilators, the label L must
vanish.
- page 42/102
In this framework, the stress energy tensor displays as expected a time
dependence due to the presence of the point-like defects.
Specifically it shows that in each defect we have a primary boundary changing
operators (whose weight depends on the monodromy) and which creates the
excited vacuum from the invariant vacuum.
This is by all means an excited spin field.
Finally (and definitely fascinating), the stress energy tensor obeys the
canonical OPE, that is the theory is still conformal (even though there is a
time dependence).
- page 43/102
In formulae, the excited vacuum used in computations is thus created by a
radially ordered product of excited spin fields hidden in the defects.
- page 44/102
We are therefore in a position to compute the correlators involving such spin
fields (however since we cannot compute the normalisation, we can compute
only quantities not involving it).
For instance we reproduce the known result of bosonization.
Moreover, since we have complete control over the algebra of the fermionic
fields, we can also compute any correlator involving both spin and matter
fields.
- page 45/102
We therefore showed that semi-phenomenological models need the ability to
compute correlators involving twist and spin fields.
We then introduced a framework to compute the instanton contribution to the
correlators using intersecting D-branes and we showed how to compute
correlators in the fermionic case involving spin fields as point-like defects
on the string worldsheet.
The question would now be how to extend this to non Abelian spin fields and,
most importantly, to twist fields, where there is no framework such as
bosonization.
- page 46/102
After considering defects and singular points in particle physics, I will
deal with time dependent singularities in cosmology.
- page 47/102
The reason is that as string theory is considered a theory of everything, its
phenomenological description should in fact include both strong and
electroweak forces as well as gravity.
- page 48/102
In particular from the gravity side, we would like to have a better view of
the cosmological implications in string theory.
- page 49/102
For instance we could try to study Big Bang models to gain some better
insight with respect to field theory.
- page 50/102
For this, one way would be to build toy models of singularities in time, in
which the singular point exists in one specific moment, rather than place.
- page 51/102
A simple way to make it so is to build toy models from time-dependent
orbifolds which can model singularities as their fixed points.
- page 52/102
In the past people already dealt with such problem finding divergences in the
computation of amplitudes.
The presence of such divergences in N-point correlators is however usually
associated to a gravitational backreaction due to exchange of gravitons.
- page 53/102
However the 4-tachyon amplitude in string theory is divergent already in the
open string sector at tree level: in other words genuine gauge theories.
The effective field theory interpretation would be a 4-point interaction of
scalar fields (higher spins would only spoil the behaviour).
- page 54/102
To investigate further, we consider the so called Null Boost Orbifold.
The construction starts from D-dimensional Minkowski spacetime through a
change of coordinates.
- page 55/102
The orbifold is then built through the periodic identification of one
coordinate along the direction of its Killing vector, which leads to momentum
quantization.
- page 56/102
From these identifications, we can build scalar wave functions obeying the
standard equations of motion.
Notice the behaviour in the time direction u which already takes a peculiar
form and the presence of the quantized momentum in a strategic place.
- page 57/102
In order to introduce the divergence problem we first consider a theory of
scalar QED.
- page 58/102
When computing the interactions between the fields, the terms involved are
entirely defined by two main integrals.
It might not be immediately visible, but given the behaviour of the scalar
functions, any vertex interaction with more than 3 fields diverges.
- page 59/102
The reason for the divergence is connected to the "strategically placed"
quantized momentum.
When when all quantized momenta vanish, in the limit of small u (that is near
the singularity) the integrands develop isolated zeros preventing the
convergence.
In fact, in this case, even a distributional interpretation (not unlike the
derivative of a delta function) fails.
- page 60/102
So far the situation is therefore somewhat troublesome.
Moreover, obvious ways to regularise the theory do not work: for instance
adding a Wilson line does not cure the problem as divergences also involve
neutral strings which would not feel the regularisation.
In fact the problems seem to arise from the vanishing volume in phase space
along the compact direction: the issue looks like geometrical, rather than
strictly gravitational.
- page 61/102
Since the field theory fails to give a reasonable value for amplitudes
involving time-dependent singularities, we could therefore ask whether string
theory can shed some light.
- page 62/102
The relevant divergent integrals are in fact present also in string theory.
They arise from interactions of massive vertices like what is shown here.
These vertices are usually overlooked as they do not play in general a
relevant role at low energy.
However it is possible that near the singularity they might actually give
their contribution.
These vertices are involved at low energy in the definition of contact terms
in the effective field theory, which therefore does not account for them.
- page 63/102
In this sense even string theory cannot give a solution to the problem.
In other words since the effective theory does not even exist, its high
energy completion is not capable of providing a better description.
- page 64/102
There is however one geometric way to escape this.
Since the issues are related to a vanishing phase space volume, analytically
speaking it is sufficient to add a non compact direction to the orbifold in
which the particle is "free to escape".
- page 65/102
While the Generalised Null Boost Orbifold has basically the same definition
through one of its Killing vector, the presence of the additional direction
acts in a different way on the definition of the scalar functions.
As you can see the new time behaviour ensures better convergence properties,
and the presence of the continuous momentum ensures that no isolated zeros
are present at any time.
Even in the worst case scenario, the arising amplitudes would still have a
distributional interpretation.
- page 66/102
We therefore showed that divergences in the simplest theories are present
both in field theory and string theory and that in the presence of
singularities, the string massive states start to play a role.
The nature of the divergences is however due to vanishing volumes in phase
space and cannot be classified as simply a gravitational backreaction.
In fact the introduction of "escape routes" for fields grants a
distributional interpretation of the amplitudes.
It is also possible to show that this is not restricted to "null boost" types
of orbifolds, but even other kinds of orbifolds present the same issues.
- page 67/102
In summary we showed that the divergences cannot be regarded as simply
gravitational, but even gauge theories (that is the open sector of the string
theory) present issues.
Their nature is however subtle and connected to the interaction of string
massive modes (or contact terms in the low energy formulation) which are not
usually taken into account.
- page 68/102
We finally move to the last section.
After the analysis of semi-phenomenological analytical models, we now
consider a computational task related to compactifications of
extra-dimensions using machine learning.
- page 69/102
We focus on Calabi-Yau manifolds in three complex dimensions.
Due to their properties and their symmetries, the relevant topological
invariants are two Hodge numbers.
As the number of possible compact Calabi-Yau 3-folds is a huge, we focus on a
subset.
- page 70/102
Specifically we focus on manifolds built as intersections of hypersurfaces in
projective spaces, that is intersections of several homogeneous equations in
the complex coordinates of the manifold.
As we are interested in studying these manifolds as topological spaces, for
each equation and projective space we do not care about the coefficients, but
only the exponents, or better the degree of the equation in a given
coordinate.
- page 71/102
The intersections can be generalised to multiple projective spaces and
equations and the manifold can be characterised by a matrix containing the
powers of the coordinates in each equation.
The problem in which we are interested is therefore to be able to take the so
called "configuration matrix" of the manifolds and predict the value of the
Hodge numbers.
- page 72/102
The real issue is now how to treat the configuration matrix and how to build
such map.
- page 73/102
We use a machine learning approach.
In very simple words it means that we want to find a new representation of
the input (possibly parametrized by some weights which we can tune and
control) such that the predicted Hodge numbers are as close as possible to
the correct result.
In this sense the machine has to learn some way to transform the input to get
a result close to what in the computer science literature is called the
"ground truth".
The measure of proximity or distance from the true value is called "loss
function" or "Lagrangian function" (with a slight abuse of naming
conventions).
The machine then learns some way to minimise this function (for instance
using gradient descent methods and updating the parameters).
- page 74/102
We thus exchange the difficult problem of finding an analytical solution with
an optimisation problem (it does not imply "easy", but it is at least
doable).
- page 75/102
In order to learn the best way of doing this, we can rely on a vast computer
science literature and use large physics datasets containing lots of samples
from which to infer a structure.
- page 76/102
In this sense the approach can merge techniques from physics, mathematics and
computer science benefiting from advancements in all fields.
- page 77/102
The approach can furthermore provide a good way to analyse data and infer
structure and advance hypothesis, which could end up overlooked using
traditional brute force algorithms.
In this case we focus on the prediction of two Hodge numbers with very
different distributions and ranges.
The data we consider were computed using top of the class computing power at
CERN in the 80s, with a huge effort by the string theory community.
In this sense Complete Intersection Calabi-Yau manifolds are a good starting
point to investigate the application of machine learning techniques because
they are well studied and characterised.
- page 78/102
The dataset we use contains less than 10000 manifolds (in machine learning
terms it is still small).
From these we remove product spaces (recognisable by their block diagonal
form of the configuration matrix) and we remove very high values of the Hodge
numbers to avoid learning "extremal configurations".
Mind that we only remove them from the training data which the machine
actually uses to learn.
In this sense we are simply not giving the machine "extremal" configurations
in an attempt to push as far as possible the application: should the machine
learn a good representation, it will automatically be capable of learning
also those configurations without a human manually feeding them.
We then define three separate folds: the largest contains training data used
by the machine to adjust the parametrisation, 10% of the data is then used
for intermediate evaluation of the process, while the last subset is used to
give the final predictions.
Differently from the validation set, the test set has not been seen by the
machine and therefore can reliably test the generalisation ability of the
algorithm.
Differently from some previous approaches we consider this as a regression
task in the attempt to let the machine learn a true map between the
configuration matrix and the Hodge numbers (in case we can also discuss the
classification approach as it has some interesting applications itself).
- page 79/102
The pruned distribution of the Hodge numbers therefore presents less outliers
than the initial dataset, but as you can see we expect the result to be
similar even without the procedure, since the number of outliers removed is
small.
This first analysis however proved to be a good success to get higher
results.
- page 80/102
The pipeline we adopt is the same used at industrial level by companies and
data scientists.
We in fact heavily rely on data analysis to improve as much as possible the
output.
- page 81/102
This for instance can be done by including additional information with
respect to the configuration matrix, that is by feeding the machine variables
which can be manually derived: by definition they are redundant but can be
used to easily learn a pattern.
In fact as we can see most of the features such as the number of projective
spaces or the number of equations in the matrix are heavily correlated with
the Hodge numbers.
Moreover using algorithms to produce a ranking of the variables such as
decision trees shows that such "engineered features" are much more important
than the configuration matrix itself.
- page 82/102
Using the "engineered data", we now get to the choice of the algorithm.
There is no general rule for this, even though there might be good guidelines
to follow.
- page 83/102
Though the approach is clearly "supervised" in the sense that the machine
learns by approximating a known result, we also tried other approaches in an
attempt to generate additional information which the machine could use.
The first approach is a clustering algorithm, intuitively used to look for a
notion of "proximity" between the configuration matrices.
This however did not play a role in the analysis.
The other is definitely more interesting and it consists in finding a better
representation of the configuration matrix using less components.
The idea is therefore to "squeeze" or "concentrate" the information in a
lower dimensional space (matrices in our case have 180 components, so we are
trying to aim for something less than that).
- page 84/102
For the predictions we first relied on traditional regression algorithms,
such as linear models, support vector machines and boosted decision trees.
I will not enter into the details and differences between the algorithms, but
we can indeed discuss them.
- page 85/102
Let me however say a few words a dimensionality reduction procedure known as
"principal components analysis" (or PCA for short), since this proved to be
an important step in the analysis.
Suppose that we have a rectangular matrix (which could be number of samples
in the dataset times the number of components of the matrix once it has been
flattened).
The idea of PCA is to project the data onto a lower dimensional surface where
the variance if maximised in order to retain as much information as possible.
This is usually used to isolate a signal from a noisy background.
Thus by isolating only the meaningful components of the matrix we can hope to
help the algorithm.
- page 86/102
Visually PCA is used to isolate the eigenvalues and eigenvectors of the
covariance matrix (or the singular values of the matrix) which do not belong
to the background.
From random matrix theory we know that the eigenvalues of an independently
and identically distributed matrix (a Wishart matrix) follow a
Marchenko-Pastur distribution.
Such matrix containing a signal would therefore be recognised by the presence
of eigenvalues outside this probability distribution.
We could therefore simply keep the corresponding eigenvectors.
In our case this resulted in an improvement of the accuracy, obtained by
retaining less than half of the components of the matrix (corresponding to
99% of the variance of the initial set).
- page 87/102
As we can see we used several algorithms to evaluate the procedure.
Previous approaches in the literature mainly relied on the direct application
of algorithms to the configuration matrix.
We extended this beyond the previously considered algorithms (mainly support
vectors) to decision trees and linear models for comparison.
- page 88/102
Techniques such as feature engineering and PCA provide a huge improvement
(even with less training data).
Let me for instance point out the fact the even a simple linear regression
reaches the same level of accuracy previously reached by more complex
algorithms, even with much less training data.
This ultimately can cut computational costs and complexity.
- page 89/102
However this does not conclude the landscape of algorithms used in machine
learning.
In fact we also used neural networks architectures.
They are a class of function approximators which use (some variants of)
gradient descent to optimise the weights.
Their layered structure is key to learn highly non linear and complicated
functions.
We focused on two distinct architectures.
The older fully connected networks were employed in previous attempts at
predicting the Hodge numbers.
They rely on a series of matrix operations to create new outputs from
previous layers.
In this sense the matrix W and the bias term b are the weights which need to
be updated.
Each node is connected to all the outputs, hence the name fully connected or
densely connected.
Equivalently this means that the matrix W does not have vanishing
entries.
To learn non linear functions this is however not sufficient: an iterated
application of these linear maps would simply result in a linear function to
be learned.
We "break" linearity by introducing an "activation function" at each layer.
The second architecture is called convolutional from its iterated application
of "sliding window function" (that is convolutions) applied on the layers.
- page 90/102
Convolutional networks have several advantages over a fully connected
approach.
Since the input in this case does not need to flattened, convolutions retain
the notion of vicinity between cells in a grid (here we have an example of a
configuration matrix as seen by a convolutional neural network).
Since they do not have one weight for each connection, they have a smaller
number of parameters to update (proportional to the size of the window).
In our specific case we cut by more than one order of magnitude the number
of parameters used with respect to fully connected networks.
Moreover weights are shared by adjacent cells, meaning that if there is a
structure to be inferred, this is the way to go to exploit the "artificial
intelligence" underlying the operations involved.
- page 91/102
In this sense a convolutional architecture can isolate defining features of
the output and pass them to the following layer as in the animation.
For instance, using a computer science analogy, this can be used to classify
objects given a picture: a convolutional neural network is literally capable
of isolating what makes a dog a dog and what distinguishes it from a cat
(even more specific it can separate a Labrador from a Golden Retriever).
- page 92/102
This has in fact been used in computer vision tasks in recent year for
pattern recognitions, object detections and spatial awareness tasks (for
instance to isolate the foreground from the background).
In this sense this is the closest approximation of artificial intelligence in
supervised tasks.
- page 93/102
My contribution in this sense is inspired by deep learning research at
Google.
In recent years they were able to devise new architectures using so called
"inception modules" in which different convolution operations are used
concurrently.
The architecture has better generalisation properties since more features can
be detected and processed at the same time.
- page 94/102
In our case we decided to go for two concurrent convolutions one scanning
each equation (the vertical kernel) of the configuration matrix, while a
second convolutions scans the projective spaces (in horizontal).
The layer structure is then concatenated until a single output is produced
(the Hodge number that is).
The idea is that this way the network can learn a relation between projective
spaces and equations and recombine them to find a new representation.
- page 95/102
As we can see even the simple introduction of a traditional convolutional
kernel (the result shown was reached with a 5x5 kernel function) is
sufficient to boost the accuracy of the predictions (previous best results in
2018 reached only 77% of accuracy on h^{1,1}).
- page 96/102
The introduction of the Inception architecture has major advantages: it uses
even less parameters than "traditional" convolutional networks, it boosts the
performance reaching near perfect accuracy, it needs a lot less data (even
with just 30% of the data for training, the accuracy is already near
perfect).
Moreover with this architecture we were able to predict also h^{2,1} with 50%
accuracy: even if does not look a reliable method to predict it (I agree, for
now), mind that previous attempts have usually avoided computing it, or they
reached accuracies as high as 8-9% (even feature engineering could boost it
only around 35%).
The network is also solid enough to predict both Hodge numbers at the same
time: trading a bit of the accuracy for a simpler model, it is in fact
possible to let the machine learn the existing relation between the Hodge
numbers without specifically inputing anything (for instance by inserting the
fact that the difference of the Hodge numbers is the Euler characteristic).
For more specific info I invite you to take a look at Harold's talk on the
subject at the recent "string_data" workshop.
- page 97/102
Deep learning can therefore be used as a predictive method, provided that one
is able to analyse the data (no black boxes should ever be admitted).
- page 98/102
It can also be used a source of inspiration for inquiries and investigations
always provided a good analysis is done beforehand.
- page 99/102
Deep learning can also be used for generalisation of patterns and relations.
As always only after careful consideration.
- page 100/102
Moreover convolutional networks look promising and with a lot of unexplored
potential.
This is in fact the first time in which they have been successfully used in
theoretical physics.
Finally, this is an interdisciplinary approach in which a lot is yet to be
learned from different perspective.
- page 101/102
More directions to investigate now remain.
In fact one could in principle exploit freedom in representing the
configuration matrices to learn the best possible representation.
Otherwise one could start to think about this in a mathematical embedding and
study what happens for CICY 4-folds (almost one million complete
intersections).
Moreover, as I was saying, this could be used as an attempt to study formal
aspects of deep learning, or even more to directly dive into the "real
artificial intelligence" and start to study the problem in a reinforcement
learning environment where the machine automatically learns a task without
knowing the final result.
- page 102/102
I will therefore leave the open question as to whether this is actually going
to be the end or just the start of something else.
In the meantime I thank you for your attention.
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