Low-power multi-mode fiber projector overcomes shallow neural networks classifiers Daniele Ancora135Matteo Negri13 Antonio Gianfrate2 Dimitris Trypogeorgos2 Lorenzo Dominici2 Daniele Sanvitto2 Federico Ricci-Tersenghi134 and Luca Leuzzi31

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Low-power multi-mode ﬁber projector overcomes shallow neural networks classiﬁers

Daniele Ancora1,3,5,∗Matteo Negri1,3, Antonio Gianfrate2, Dimitris Trypogeorgos2,

Lorenzo Dominici2, Daniele Sanvitto2, Federico Ricci-Tersenghi1,3,4, and Luca Leuzzi3,1

1Department of Physics, Universit`a di Roma la Sapienza, Piazzale Aldo Moro 5, I-00185 Rome, Italy

2Institute of Nanotechnology, Consiglio Nazionale delle Ricerche (CNR-NANOTEC), Via Monteroni, I-73100 Lecce, Italy

3Institute of Nanotechnology, Soft and Living Matter Laboratory,

Consiglio Nazionale delle Ricerche (CNR-NANOTEC), Piazzale Aldo Moro 5, I-00185 Rome, Italy

4Istituto Nazionale di Fisica Nucleare, sezione di Roma1, Piazzale Aldo Moro 5, I-00185 Rome, Italy and

5Epigenetics and Neurobiology Unit, European Molecular Biology

Laboratory (EMBL Rome), Via Ramarini 32, 00015 Monterotondo, Italy

(Dated: December 18, 2023)

In the domain of disordered photonics, the characterization of optically opaque materials for

light manipulation and imaging is a primary aim. Among various complex devices, multi-mode

optical ﬁbers stand out as cost-eﬀective and easy-to-handle tools, making them attractive for several

tasks. In this context, we cast these ﬁbers into random hardware projectors, transforming an input

dataset into a higher dimensional speckled image set. The goal of our study is to demonstrate that

using such randomized data for classiﬁcation by training a single logistic regression layer improves

accuracy compared to training on direct raw images. Interestingly, we found that the classiﬁcation

accuracy achieved is higher than that obtained with the standard transmission matrix model, a

widely accepted tool for describing light transmission through disordered devices. We conjecture

that the reason for such improved performance could be due to the fact that the hardware classiﬁer

operates in a ﬂatter region of the loss landscape when trained on ﬁber data, which aligns with the

current theory of deep neural networks. These ﬁndings suggest that the class of random projections

operated by multi-mode ﬁbers generalize better to previously unseen data, positioning them as

promising tools for optically-assisted neural networks. With this study, in fact, we want to contribute

to advancing the knowledge and practical utilization of these versatile instruments, which may play

a signiﬁcant role in shaping the future of neuromorphic machine learning.

Keywords: Multi-mode ﬁbers, random projections, disordered photonics, neural networks,

classiﬁcation, MNIST.

I. INTRODUCTION

A sound understanding of the enormous success of

Neural Networks (NNs) in learning processes and infer-

ence tasks is still lacking. The fundamental point is to

understand why such architectures, which can have even

billions of parameters, do not severely overﬁt data, as

predicted by statistical learning theory and the so-called

bias-variance tradeoﬀ (see for example [1] or [2]). The

abundance of learnable parameters, in fact, is arguably

the most universal feature in the zoo of NN architectures.

Interestingly, it is known that, given a chosen NN archi-

tecture, most of the model parameters adapt little-to-

nothing during the learning procedure [3, 4], suggesting

that random projections may play an equally important

role in NNs. Recent works, in fact, have shown that it

is possible to train a simple two-layer model by learning

only the upper layer, interpreting the ﬁrst one as a ran-

dom projection [5, 6]. These results were strengthened

further by Baldassi et al. [7], who proved that increas-

ing the dimension of the random projection leads to the

production of wide and ﬂat regions in the loss landscape

(the function that is minimized during the training of

the model), which are related to the good generalization

∗daniele.ancora@uniroma1.it; daniele.ancora@cnr.it

properties in neural networks. The ability of a neural net-

work, that has been trained over a given dataset (train-

ing dataset), to generalize well is the ability of displaying

good performances when applied to data over which it

was not trained (test dataset). In the framework of the

loss landscape description an improvement in the gener-

alization means that models that lie in ﬂat regions make

less mistakes when they classify previously unseen data.

Finally, recent evidence is provided [8] that the way the

random projection is chosen is fundamental to determin-

ing the generalization properties of the upper layer of

these simple models. This suggests that diﬀerent classes

of random (possibly non-linear) projections impact dif-

ferently on the performance of the models.

In this context, we are interested in studying hard-

ware random projectors, such as those employed in the

ﬁeld of photonic neuromorphic computing [9, 10]. The

advantage of using optical neural networks (ONN) is

that neurons can interact by exploiting light scattering

[11–13] and photon interference [14, 15] at the speed

of light. Tools for shaping and controlling the light-

ﬁeld [16] are becoming so versatile that the ﬁeld is un-

der constant development, aiming at high-speed, high-

throughput optical-based computing architectures. All-

optical neural networks [11, 17], in particular, have the

potential to be great tools for fast computation, though

they often require an accurate modeling of the optical

arXiv:2210.04745v3 [physics.optics] 2 Apr 2025

system to perform consistent back-propagation update

[18]. However, the ﬁne-tuning of the optical parameters

is challenging due to discrepancies between the response

of the real system and the physical model employed to

describe the architecture. This reality gap often reduces

the expected performance of the network [19, 20], requir-

ing additional corrections at software-level [12], training

enforcement via hybrid strategies [18], or employing NNs

to more accurately model the optical response of the sys-

tem [20].

In this rapidly evolving scenario, the class of ran-

dom projections realized by multi-mode ﬁbers (MMF)

are promising candidates for realizing ONNs. These de-

vices scramble the photons due to scattering events oc-

curring during the light-ﬁeld propagation, yielding to the

formation of speckle patterns that are, in fact, random

projections. Although the light transmission can be re-

garded as a linear process [21] in which input modes are

coupled with output modes via a complex transmission

rule, interference takes place when dealing with the mea-

surement of the light-ﬁeld intensity. Since the detection is

nonlinear, MMFs can be used [22] to classify time-domain

waveforms (using saturation eﬀects as further nonlinear-

ity) [23], in pattern classiﬁcation of 2-bits sequences [24],

or for binary (human, not human) facial recognition [25].

Furthermore, when dealing with more complex classiﬁ-

cation tasks, high-power laser pulses were employed to

trigger the nonlinear response of the ﬁber itself [26]. Due

to the increasing interest in the employment of MMFs

as random projector computing devices, we decided to

study their behavior in carrying out classiﬁcation tasks

in a linear, low-power continuum regime. Although our

MMF-based optical neural network does not employ feed-

back, we will see how its performances in classiﬁcation are

considerable, as in reservoir computing systems [27–30].

We do this by comparing the performance of the phys-

ical neural network to that obtained with random Gaus-

sian linear projections and to that of a transmission ma-

trix approach, the model commonly used to describe light

propagation in disordered structures [21, 31]. We per-

form our study statistically, shuﬄing the training set to

assess the average behavior of the optical computing un-

der diﬀerent training and initialization conditions. Re-

markably, a single MMF provides simultaneously two in-

dependent (though deterministically linked) projections

at both edges of the ﬁber, which we study separately

using diﬀerent saturation regimes. Here, we show that

the real physical MMF leads to accuracy higher than its

corresponding transmission matrix model, highlighting

the reality gap between model theory and experimental

results. To assess the reason of this performance gap,

we study the characteristic of complex-valued random

projections in terms of ﬂatness of the local energy land-

scape, proving that the MMF projection is more robust

than those provided by alternative datasets. Addition-

ally, we characterize the behavior of a hardware-based

neural network using optical ﬁbers in terms of the num-

bers of the modes employed. We have set up our study

not for achieving the best performance in classiﬁcation

tasks, but rather to deepen the understanding of physi-

cal neural networks against their physical model, giving

insights on the usage of MMF ﬁbers for optical compu-

tation.

II. MATERIALS AND METHODS

In a low-power regime, a generic multi-mode ﬁber

transports the electromagnetic ﬁeld via a linear process

[21] so that the light propagation can be described using

a simple multiplication of the input signal by a matrix

that encodes the transmission rule:

y=Tx.(1)

In this descriptive model, xis the controlled input, Tis

the (complex-valued and typically unknown) transmis-

sion matrix of the medium, and yis the output ﬁeld.

Despite its propagation, the way we measure the MMF

output is not linear for two reasons. First, photons carry

complex signals, i.e., the electromagnetic ﬁeld associated

to each propagation mode is characterized by amplitude

and phase. Current electronic devices cannot follow the

rapid oscillation of the ﬁeld, which makes impossible the

measure of the phase information. Assuming the possi-

bility that the readout is also perturbed by an additive

noise ε, the camera only sees the noise aﬀected intensity

distribution:

|y|2=|Tx|2+ε. (2)

Second, the camera has a well deﬁned sensitivity range

that depends on each pixels capability to store intensity

change. If the signal reaching a given pixel exceeds the

sensitivity, the measure gets clipped at the peak (over-

exposure) or at the bottom (underexposure). In analogy

to machine learning terminology, the measuring process

can be described by a non-linear activation function σ(·)

that acts on the result of a complex-valued linear trans-

mission, Tx. For instance, the camera recording process

can be represented using the saturating linear transfer

function (SatLin):

σ(Tx) = min max d, |Tx|2+ε,2b−1,(3)

where the quantity dis the intensity threshold under

which the measure is not recorded, and bis the bit depth

of the camera.

These considerations make the readout of a coherent

ﬁeld non-linear, as well as its inverse transmission recov-

ery problem [21, 31–36]. Such matrix can be estimated

by using the four-phases method [21], Bayesian optimiza-

tion [37], or iterative Gerchberg-Saxton schemes [38, 39].

However, the characterization of the device in terms of

its transmission rule is not the main scope of this paper,

nor to circumvent the limitations of the measuring pro-

cess. Instead, we want to study the multi-modal random-

projection nature of the ﬁber to perform optical com-

puting. In the neural network framework, the ﬁber can

FIG. 1. Schematics of the shallow optical neural network with the MMF. Panel a), simpliﬁed scheme of light transport through

multi-mode ﬁbers. The MNIST data set, modulated by a spatial light modulator, enters the MMF on the input facet. During

propagation, the light gets scrambled with a random but deterministic process, giving rise to the speckle pattern measured in

the camera. Panel b), corresponding neural network interpretation of the light propagation scheme. The MNIST set constitutes

the input vector of a linear complex layer with static weights. The non-linear operation is determined by the camera that reads

the intensity of a complex ﬁeld. Successively, a linear classiﬁcation layer is trained using the output of the ﬁber. Panel c),

scheme of the imaging setup.

be seen as an optical analogous of a densely-connected

network composed by a single “hidden layer” with ﬁxed

weights [40]. In this shallow architecture, the MMF layer

already contains a particular realization of static weights

(the transmission matrix T), which depends upon the

physical status of the optical ﬁber. This property allows

to perform random -but deterministic- projections at the

speed of light using a ﬁxed transmission rule, which can

be read out by the camera. Given these considerations,

the MMF is a good candidate to perform non-linear opti-

cal computation using continuous laser source even using

inexpensive and large (thus easier to handle in a setup)

optical multi-mode ﬁbers. In particular, if we let just a

few modes propagate into the input facet of an MMF that

supports many more, all the output modes will be acti-

vated, implying a mapping of the kind few-to-many. In

this latter case, the optical hidden layer (i.e., MMF and

camera) can perform densely-connected random projec-

tions on a higher dimensional space.

The goal of this study is to carry out image classiﬁca-

tion by concatenating a software-trained linear layer to

the measured output of a MMF, produced by inserting a

given image from the dataset into the input edge of the

ﬁber. We choose to approach the MNIST classiﬁcation

problem in order to carry out a widely studied non-linear

task. The only parameters that we train are those of a

simple logistic regression layer, which is known to achieve

poor performances on the standard MNIST set, reaching

a maximum classiﬁcation accuracy of 92.7% [29]. Ex-

ploiting random projection provided by the MMF, an

optical device that is known to be linear, we compare

with the performances obtained using reference datasets.

In this study, we train the parameters of the logistic clas-

siﬁer using six diﬀerent input datasets:

1. Original MNIST. The standard MNIST dataset,

constituted by images of l×lpixels. The accu-

racy performance of this set is the baseline of our

study.

2. Upscaled MNIST. Each image at the original res-

olution is expanded by a factor L/l using a linear

spline interpolation to reach the target size of L×L.

3. Randomized MNIST. The MNIST set is linearly

multiplied with a Gaussian random matrix with

positive entries. This maps the dataset into a

higher-dimension space, producing images with a

side L≫lpixels.

4. MMF α-cam. The speckled output of the MMF is

recorded with a resolution of L×Lpixels. Each

speckle pattern is the result of sending a MNIST

image on the input edge of the ﬁber, recording the

output after disordered propagation. The patterns

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Low-powermulti-modefiberprojectorovercomesshallowneuralnetworksclassifiersDanieleAncora1,3,5,∗MatteoNegri1,3,AntonioGianfrate2,DimitrisTrypogeorgos2,LorenzoDominici2,DanieleSanvitto2,FedericoRicci-Tersenghi1,3,4,andLucaLeuzzi3,11DepartmentofPhysics,Universit`adiRomalaSapienza,PiazzaleAldoMoro5,I-001...

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Low-power multi-mode fiber projector overcomes shallow neural networks classifiers Daniele Ancora135Matteo Negri13 Antonio Gianfrate2 Dimitris Trypogeorgos2 Lorenzo Dominici2 Daniele Sanvitto2 Federico Ricci-Tersenghi134 and Luca Leuzzi31.pdf

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Low-power multi-mode fiber projector overcomes shallow neural networks classifiers Daniele Ancora135Matteo Negri13 Antonio Gianfrate2 Dimitris Trypogeorgos2 Lorenzo Dominici2 Daniele Sanvitto2 Federico Ricci-Tersenghi134 and Luca Leuzzi31

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