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applications are suggested, and it is clear that no existing thermal
management strategy or technology meets all these requirements., URL =
http://jes.ecsdl.org/content/158/3/R1.abstract, eprint =
http://jes.ecsdl.org/content/158/3/R1.full.pdf+html, journal = Journal of The
Electrochemical Society

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Estimation for Lithium-Ion Batteries Using Neural Networks and EKF, year=2010,
volume=57, number=12, pages=4178-4187, keywords=Kalman filters;battery
chargers;covariance matrices;neural nets;power engineering computing;secondary
cells;battery charging;battery terminal voltage;covariance matrix;extended
Kalman filter;lithium-ion batteries;neural networks;state-of-charge
estimation;Battery charge measurement;Electrochemical impedance
spectroscopy;Impedance measurement;Laboratories;Neural
networks;Permission;Pulse power systems;State
estimation;Testing;Voltage;Batteries;Kalman
filtering;estimation;monitoring;neural networks (NNs),
doi=10.1109/TIE.2010.2043035, ISSN=0278-0046, month=Dec,

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”111 - 124”, year = ”2005”, note = ””, issn = ”0378-7753”, doi =
”http://dx.doi.org/10.1016/j.jpowsour.2004.05.064”, url =
”http://www.sciencedirect.com/science/article/pii/S0378775304008596”, author =
”S.C. Chen and C.C. Wan and Y.Y. Wang”, keywords = ”Thermal modelling”,
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”Heat dissipation”, keywords = ”Thermal management ”

@articleDebert2013, title = ”An observer looks at the cell temperature in
automotive battery packs ”, journal = ”Control Engineering Practice ”, volume
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”0967-0661”, doi = ”http://dx.doi.org/10.1016/j.conengprac.2013.03.001”, url =
”http://www.sciencedirect.com/science/article/pii/S0967066113000464”, author =
”Maxime Debert and Guillaume Colin and Gérard Bloch and Yann Chamaillard”,
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”Cell temperature estimation”, keywords = ”Internal resistance estimation”,
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@ARTICLEDey2015a, author=Dey, S. and Ayalew, B. and Pisu, P., journal=Control
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State-of-Charge Estimation of Lithium-Ion Cells Based on a Reduced
Electrochemical Model, year=2015, volume=23, number=5, pages=1935-1942,
keywords=Lyapunov methods;battery management systems;control system
synthesis;linear matrix inequalities;nonlinear control
systems;observers;robust control;secondary cells;variable structure
systems;Luenberger term design;Lyapunov direct method;SOC information;battery
management systems;cell-level state-of-charge information;linear matrix
inequality;lithium-ion cells;module-level state-of-charge
information;nonlinear robust observers;reduced order electrochemical
model;state-of-charge estimation;variable structure term;Electrodes;Integrated
circuit modeling;Mathematical model;Observers;System-on-
chip;Uncertainty;Electrochemical model;lithium-ion (Li-ion)
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”http://www.sciencedirect.com/science/article/pii/S2405896315019321”, author =
”Satadru Dey and Zoleikha Abdollahi Biron and Sagar Tatipamula and Nabarun Das
and Sara Mohon and Beshah Ayalew and Pierluigi Pisu”, keywords = ”Lithium-ion
Batteries”, keywords = ”Fault Diagnosis”, keywords = ”Thermal Failure”,
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@articleDoughty2002, title = ”Lithium battery thermal models ”, journal =
”Journal of Power Sources ”, volume = ”110”, number = ”2”, pages = ”357 -
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”http://dx.doi.org/10.1016/S0378-7753(02)00198-2”, url =
”http://www.sciencedirect.com/science/article/pii/S0378775302001982”, author =
”Daniel H Doughty and Paul C Butler and Rudolph G Jungst and E.Peter Roth”,
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”Calorimeter”, keywords = ”Thermal model”, keywords = ”Electrochemical model”,
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@INPROCEEDINGSGould2008, author=Gould, C.R. and Bingham, C.M. and Stone, D.A.
and Bentley, P., booktitle=Power Electronics, Electrical Drives, Automation
and Motion, 2008. SPEEDAM 2008. International Symposium on, title=Novel
battery model of an all-electric personal rapid transit vehicle to determine
state-of-health through subspace parameter estimation and a Kalman Estimator,
year=2008, pages=1217-1222, keywords=ageing;battery management systems;battery
powered vehicles;electrical safety;equivalent circuits;parameter
estimation;rapid transit systems;real-time systems;Kalman estimator;Randles
model;all-electric personal rapid transit vehicle;battery ageing;equivalent
circuit;online monitoring;real-time adaptive battery model;safety-critical
system;subspace parameter estimation;Battery powered vehicles;Condition
monitoring;Degradation;Equivalent circuits;Fluctuations;Kalman
filters;Parameter estimation;State estimation;Temperature;Voltage;Battery
management systems;Energy storage;Parameter estimation;System identification,
doi=10.1109/SPEEDHAM.2008.4581332, month=June,

@articleGuo2010, title = ”Three-dimensional thermal finite element modeling of
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”2010”, note = ””, issn = ”0378-7753”, doi =
”http://dx.doi.org/10.1016/j.jpowsour.2009.10.090”, url =
”http://www.sciencedirect.com/science/article/pii/S0378775309019417”, author =
”Guifang Guo and Bo Long and Bo Cheng and Shiqiong Zhou and Peng Xu and
Binggang Cao”, keywords = ”Thermal model”, keywords = ”Oven abuse test”,
keywords = ”Thermal runaway”, keywords = ”Lithium-ion battery ”

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”http://www.sciencedirect.com/science/article/pii/S0378775399001780”, author =
”S. Al Hallaj and H. Maleki and J.S. Hong and J.R. Selman”, keywords =
”Thermal modeling”, keywords = ”Thermal runaway”, keywords = ”Sony cell”,
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@articleHu2012, title = ”Battery cell state-of-charge estimation using linear
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”http://dx.doi.org/10.1016/j.jpowsour.2006.09.006”, url =
”http://www.sciencedirect.com/science/article/pii/S0378775306018349”, author =
”Il-Song Kim”, keywords = ”State of charge estimation”, keywords = ”Sliding
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@ARTICLEKim2014, author=Youngki Kim and Mohan, S. and Siegel, J.B. and
Stefanopoulou, A.G. and Yi Ding, journal=Control Systems Technology, IEEE
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Cylindrical Battery Cells Under Unknown Cooling Conditions, year=2014,
volume=22, number=6, pages=2277-2286, keywords=Kalman filters;battery powered
vehicles;cooling;estimation theory;forced convection;hybrid electric
vehicles;iron compounds;lithium compounds;nonlinear filters;polynomial
approximation;secondary cells;temperature distribution;DKF;LiFePO4;battery
core temperature estimation;battery current measurement;cylindrical lithium-
iron-phosphate battery cell;dual Kalman filter;extended Kalman filter;forced-
air convective cooling;homogeneous volume;hybrid electric vehicle drive
cycle;isotropic volume;polynomial approximation;reduced-order thermal
model;surface temperature measurement;temperature distribution
estimation;unknown cooling condition;voltage measurement;volume averaged
lumped-value;Batteries;Battery charge measurement;Kalman filters;Lithium
batteries;Parameter estimation;Reduced order systems;State
estimation;Temperature distribution;Dual Kalman filter (DKF);lithium ion (Li-
ion) batteries;reduced-order model;state and parameter estimation;thermal
modeling;thermal modeling., doi=10.1109/TCST.2014.2309492, ISSN=1063-6536,
month=Nov,

@ARTICLEKlein2013, author=Klein, R. and Chaturvedi, N.A. and Christensen, J.
and Ahmed, J. and Findeisen, R. and Kojic, A., journal=Control Systems
Technology, IEEE Transactions on, title=Electrochemical Model Based Observer
Design for a Lithium-Ion Battery, year=2013, volume=21, number=2,
pages=289-301, keywords=algebra;battery powered vehicles;electric current
measurement;observers;partial differential equations;safety;secondary
cells;voltage measurement;Li;current measurements;electric-vehicle drive
cycles;electrochemical model based observer design;energy storage;internal
temperature;lithium-ion battery;mobile electrification;model based
controller;optimal operation;output error injection observer;partial
differential-algebraic equations;safe operation;safety;state estimation
strategy;unmodeled dynamics;voltage
measurements;Batteries;Electrodes;Lithium;Mathematical
model;Observers;Solids;Battery management systems;PDE observer
design;electrochemical model;lithium ion batteries,
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@ARTICLELin2013, author=Xinfan Lin and Perez, H.E. and Siegel, J.B. and
Stefanopoulou, A.G. and Yonghua Li and Anderson, R.D. and Yi Ding and
Castanier, M.P., journal=Control Systems Technology, IEEE Transactions on,
title=Online Parameterization of Lumped Thermal Dynamics in Cylindrical
Lithium Ion Batteries for Core Temperature Estimation and Health Monitoring,
year=2013, volume=21, number=5, pages=1745-1755, keywords=condition
monitoring;lithium;observers;parameter estimation;secondary cells;temperature
measurement;26650 lithium iron phosphate-graphite;Li;adaptive observer;core
temperature estimation;cylindrical lithium ion batteries;health
monitoring;lumped thermal dynamics;online parameter identification;online
parameterization;online parameterization methodology;overheating;surface
temperature measurement;temperature-dependent internal resistance;thermal
model;thermal monitoring;Adaptation models;Batteries;Battery charge
measurement;Coolants;Heating;Resistance;Temperature measurement;Adaptive
estimation;core temperature;lithium ion battery;state of health,
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@INPROCEEDINGSLombardi2014, author=Lombardi, W. and Zarudniev, M. and Lesecq,
S. and Bacquet, S., booktitle=Control Conference (ECC), 2014 European,
title=Sensors fault diagnosis for a BMS, year=2014, pages=952-957,
keywords=battery management systems;fault diagnosis;sensors;BMS;battery
management system;battery pack design;cells;current;sensor faults;sensors
fault diagnosis;temperature;voltage;Batteries;Equations;Mathematical
model;Temperature measurement;Temperature sensors;Voltage measurement,
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@articlePlett2004, title = ”Extended Kalman filtering for battery management
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”2”, pages = ”277 - 292”, year = ”2004”, note = ””, issn = ”0378-7753”, doi =
”http://dx.doi.org/10.1016/j.jpowsour.2004.02.033”, url =
”http://www.sciencedirect.com/science/article/pii/S0378775304003611”, author =
”Gregory L. Plett”, keywords = ”Battery management system (BMS)”, keywords =
”Hybrid-electric-vehicle (HEV)”, keywords = ”Extended Kalman filter (EKF)”,
keywords = ”State-of-charge (SOC)”, keywords = ”State-of-health (SOH)”,
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@articleRichardson2014, title = ”Battery internal temperature estimation by
combined impedance and surface temperature measurement ”, journal = ”Journal
of Power Sources ”, volume = ”265”, number = ””, pages = ”254 - 261”, year =
”2014”, note = ””, issn = ”0378-7753”, doi =
”http://dx.doi.org/10.1016/j.jpowsour.2014.04.129”, url =
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”Robert R. Richardson and Peter T. Ireland and David A. Howey”, keywords =
”Lithium-ion”, keywords = ”Electrochemical impedance spectroscopy”, keywords =
”Temperature”, keywords = ”Thermal runaway”, keywords = ”Battery management
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@INPROCEEDINGSSingh2013, author=Singh, A. and Izadian, A. and Anwar, S.,
booktitle=Industrial Electronics Society, IECON 2013 - 39th Annual Conference
of the IEEE, title=Fault diagnosis of Li-Ion batteries using multiple-model
adaptive estimation, year=2013, pages=3524-3529, keywords=Kalman
filters;electrochemical impedance spectroscopy;equivalent circuits;fault
diagnosis;secondary cells;Kalman filters;battery fault detection unit;battery
models;equivalent circuit;fault diagnosis;impedance spectroscopy;model
parameter variation;multiple model adaptive estimation;secondary
batteries;Adaptation models;Batteries;Circuit faults;Impedance;Integrated
circuit modeling;Kalman filters;Mathematical model,
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@ARTICLEZhang2002, author=Xiaodong Zhang and Polycarpou, M.M. and Parisini,
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diagnosis;nonlinear control systems;nonlinear dynamical systems;robust
control;state estimation;uncertain systems;abrupt faults;adaptive residual
threshold;fault isolability conditions;incipient faults;isolation
estimators;nonlinear adaptive estimator;nonlinear uncertain dynamic
systems;online health monitoring;robust detection and isolation scheme;robust
fault diagnosis;simulation examples;Algorithm design and
analysis;Equations;Fault detection;Fault diagnosis;Monitoring;Nonlinear
systems;Robustness;State estimation;Uncertain systems;Uncertainty,
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@ARTICLEDey2015c, author = S. Dey and B. Ayalew and P. Pisu, title = Nonlinear
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@ARTICLEMoura2014, title=Adaptive partial differential equation observer for
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number=1, pages=011015, year=2014, publisher=American Society of Mechanical
Engineers

@articleDey2017acc, author = S. Dey and H. E. Perez and Scott Moura, title =
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@INPROCEEDINGSDey2015e, author = S. Dey and B. Ayalew, title = A Diagnostic
Scheme for Detection, Isolation and Estimation of Electrochemical Faults in
Lithium-Ion Cells, booktitle = 2014 ASME Dynamic Systems and Control
Conference (DSCC), year = 2014, editor = , volume = 4, series = 5, pages =
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a lithium-ion battery management system, author=Marcicki, Jim and Onori,
Simona and Rizzoni, Giorgio, booktitle=ASME 2010 Dynamic Systems and Control
Conference, pages=607–614, year=2010, organization=American Society of
Mechanical Engineers

@inproceedingsliu2014, title=Fault detection and isolation for lithium-ion
battery system using structural analysis and sequential residual generation,
author=Liu, Zhentong and Ahmed, Qadeer and Rizzoni, Giorgio and He, Hongwen,
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G. Stefanopoulou and Yi Ding and Matthew P. Castanier”, keywords = ”Lithium
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”Satadru Dey and Zoleikha Abdollahi Biron and Sagar Tatipamula and Nabarun Das
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@ARTICLEMotapon2016, author=S. N. Motapon and A. Lupien-Bedard and L. A.
Dessaint and H. Fortin-Blanchette and K. Al-Haddad, journal=IEEE Transactions
on Industrial Electronics, title=A Generic Electro-Thermal Li-Ion Battery
Model for Rapid Evaluation of Cell Temperature Temporal Evolution, year=2016,
volume=PP, number=99, pages=1-1, keywords=Batteries;Data models;Discharges
(electric);Integrated circuit modeling;Mathematical model;Thermal
resistance;CC-CV charging;Li-ion batteries;electrical simulation;electro-
thermal modelling;generic battery model;least-square algorithm,
doi=10.1109/TIE.2016.2618363, ISSN=0278-0046, month=,

@ARTICLEZhang2016tie, author=C. Zhang and K. Li and J. Deng and S. Song,
journal=IEEE Transactions on Industrial Electronics, title=Improved Real-time
State-of-Charge Estimation of LiFePO4 Battery Based on a Novel Thermoelectric
Model, year=2016, volume=PP, number=99, pages=1-1,
keywords=Batteries;Estimation;Heating;Integrated circuit modeling;State of
charge;Temperature;Temperature measurement;Internal temperature estimation;SOC
estimation;joint extended Kalman Filter;thermoelectric model,
doi=10.1109/TIE.2016.2610398, ISSN=0278-0046, month=,

@ARTICLEMejdoubi2016, author=A. El Mejdoubi and A. Oukaour and H. Chaoui and
H. Gualous and J. Sabor and Y. Slamani, journal=IEEE Transactions on
Industrial Electronics, title=State-of-Charge and State-of-Health Lithium-Ion
Batteries Diagnosis According to Surface Temperature Variation, year=2016,
volume=63, number=4, pages=2391-2402, keywords=Kalman filters;Lyapunov
methods;nonlinear filters;secondary cells;EKF;Lyapunov direct
method;SOC;SOH;STV;extended Kalman filter;lithium-ion batteries;state-of-
charge;state-of-health estimation technique;surface temperature
variation;Aging;Batteries;Discharges (electric);Estimation;Kalman
filters;Resistance;Temperature measurement;Adaptive observer;Extended Kalman
Filter;Lithium-ion batteries;Lyapunov stability;Parameters estimation;State-
of-Charge;State-of-Health;extended Kalman filter (EKF);lithium-ion
batteries;parameters estimation;state-of-charge (SOC);state-of-health (SOH),
doi=10.1109/TIE.2015.2509916, ISSN=0278-0046, month=April,

@ARTICLESidhu2015tie, author=A. Sidhu and A. Izadian and S. Anwar,
journal=IEEE Transactions on Industrial Electronics, title=Adaptive Nonlinear
Model-Based Fault Diagnosis of Li-Ion Batteries, year=2015, volume=62,
number=2, pages=1002-1011, keywords=equivalent circuits;fault diagnosis;iron
compounds;lithium compounds;nonlinear filters;phosphorus compounds;secondary
cells;Li-ion batteries;LiFePO4;equivalent circuit methodology;extended Kalman
filters;fault diagnosis technique;impedance spectroscopy;multiple-model
adaptive estimation technique;nonlinear battery signature-fault
models;parameter variation;residual signals;terminal voltage;Adaptation
models;Batteries;Circuit faults;Equivalent circuits;Fault
diagnosis;Impedance;Integrated circuit modeling;Extended Kalman filter
(EKF);Li-ion battery;fault diagnosis;multiple-model adaptive fault diagnosis,
doi=10.1109/TIE.2014.2336599, ISSN=0278-0046, month=Feb,

@ARTICLERahimi2014tie, author=H. Rahimi-Eichi and F. Baronti and M. Y. Chow,
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Parameter Identification and State-of-Charge Coestimation for Lithium-Polymer
Battery Cells, year=2014, volume=61, number=4, pages=2053-2061, keywords=RC
circuits;battery charge measurement;estimation theory;least squares
approximations;parameter estimation;piecewise linear techniques;secondary
cells;RC equivalent circuit;RC parameters;SOC coestimation method;battery
dynamics;charging-discharging rates;estimation algorithm;lithium-polymer
batteries;moving window least squares parameter-identification
technique;nonupdating parameters;open-circuit voltage;piecewise linear
approximation with;plug-in hybrid electric vehicles;real-time
estimation;resistance-capacitance equivalent circuit;smart grid
applications;state of charge;Battery modeling;observer;open-circuit
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# Adaptive Point Transformer

Alessandro Baiocchi  Indro Spinelli  Alessandro Nicolosi  Simone Scardapane

###### Abstract

The recent surge in 3D data acquisition has spurred the development of
geometric deep learning models for point cloud processing, boosted by the
remarkable success of transformers in natural language processing. While point
cloud transformers (PTs) have achieved impressive results recently, their
quadratic scaling with respect to the point cloud size poses a significant
scalability challenge for real-world applications. To address this issue, we
propose the Adaptive Point Cloud Transformer (AdaPT), a standard PT model
augmented by an adaptive token selection mechanism. AdaPT dynamically reduces
the number of tokens during inference, enabling efficient processing of large
point clouds. Furthermore, we introduce a budget mechanism to flexibly adjust
the computational cost of the model at inference time without the need for
retraining or fine-tuning separate models. Our extensive experimental
evaluation on point cloud classification tasks demonstrates that AdaPT
significantly reduces computational complexity while maintaining competitive
accuracy compared to standard PTs. The code for AdaPT is made publicly
available.

###### keywords:

Transformer , Geometric Deep Learning , Point Clouds , Gumbel-Softmax , Token
Selection

††journal: Neural Networks

[inst1]organization=Sapienza University of Rome, Department of Computer,
Control and Management Engineering,addressline=Via Ariosto 25, city=Rome,
postcode=00185, country=Italy

[inst3]organization=Sapienza University of Rome, Department of Computer
Science,addressline=via Salaria, 113, city=Rome, postcode=00198, country=Italy

[inst4]organization=Leonardo Labs,addressline=Piazza Monte Grappa, 4,
city=Rome, postcode=00195, country=Italy

[inst2]organization=Sapienza University of Rome, Department of Electronics and
Telecommunication Engineering,addressline=via Eudossiana, 18, city=Rome,
postcode=00184, country=Italy

##  1 Introduction

The increasing ubiquity of point cloud data across diverse domains, from
autonomous navigation to environmental monitoring, has spurred significant
advancements in point cloud representation learning (Camuffo et al., 2022).
Unlike traditional two-dimensional data, point clouds exhibit unique
characteristics, including their inherent invariance under the permutation of
their constituent points. These properties pose inherent challenges for
conventional neural network architectures, necessitating the development of
tailored methodologies specifically designed for point cloud processing.

Early attempts to tackle point cloud analysis relied on localized message-
passing approaches inspired by graph neural networks (Kipf and Welling, 2017),
as exemplified by PointNet (Charles et al., 2017) and its extensions (Qi et
al., 2017; Guo et al., 2020). However, these methods often suffer from limited
receptive field sizes, particularly when dealing with large-scale point
clouds.

In recent years, the adoption of transformers (Vaswani et al., 2017),
initially introduced for natural language processing, has emerged as a
promising alternative for point cloud representation learning (Guo et al.,
2021). Transformers excel in handling unordered data through their attention
mechanism, which operates on tokenized representations of the input data. This
tokenization enables transformers to readily adapt to diverse data types,
treating each token as an independent entity, regardless of its origin. After
the tokenization, the self-attention mechanism is crucial in preserving the
permutation invariance of point clouds. By operating on an unordered set of
tokens, self-attention allows transformers to learn relationships between
points without explicitly considering their order. However, the quadratic
computational complexity of self-attention poses a significant challenge for
scaling transformers to large point clouds (Hackel et al., 2017). Several
approaches have been proposed to mitigate this issue. One strategy involves
using a fixed number of tokens by grouping multiple points into single tokens
(Guo et al., 2021). This method offers computational efficiency but sacrifices
accuracy due to the aggregation of heterogeneous points. Another approach
involves knowledge distillation, where a large, pre-trained model (teacher) is
used to train a smaller, more efficient model (student) (Zhang et al., 2023;
Yao et al., 2022). This approach can reduce inference time at the cost of
reduced accuracy.

Token selection, another technique for reducing token count, involves
selectively eliminating tokens based on a relevance criteria. This method has
been successfully employed in computer vision applications, such as AdaViT
(Meng et al., 2022) and DynamicViT (Rao et al., 2021), which dynamically
assess token importance for the main tasks. Yang et al. (Yang et al., 2019)
adapted this framework for point cloud preprocessing. However, these methods
require users to pre-define a token budget, limiting flexibility.

To address these shortcomings, we propose the Adaptive Point Cloud Transformer
(AdaPT), a novel model that seamlessly integrates adaptive downsampling and
dynamic resource allocation. AdaPT adaptively selects the number of tokens to
retain based on an input parameter specified at inference time, enabling a
flexible trade-off between computational resources and prediction accuracy.
AdaPT’s ability to adjust its computational budget based on available
resources makes it well-suited for real-world applications where resource
constraints necessitate efficient processing with variable inference time,
especially when dealing with variable-sized point clouds. The proposed AdaPT
model represents a significant step in developing scalable and adaptable point
cloud representation learning methods.

#### Organization of the paper

After introducing AdaPT in Section 3, we demonstrate its effectiveness through
extensive experiments in Section 4. In Section 4.4 we provide an in-depth
analysis of AdaPT’s performance and several ablation studies to validate its
performance in handling large-scale point clouds with varying computational
budgets. We conclude in 5 with some final remarks and possible future
directions.

##  2 Related Works

In this section, we present related works on two aspects: Point Cloud
processing and Transformer-based models.

###  2.1 Point Cloud Representation and Processing

PointNet is considered the pioneering work in point cloud learning (Charles et
al., 2017), followed by PointNet++ (Qi et al., 2017). These models directly
process point cloud data and learn a representation of the input using a feed-
forward network, which is applied independently to each point. The information
from all points is then aggregated in a permutation-invariant manner.
PointNet++ enhances this approach by introducing local features obtained by
aggregating information from neighboring points through ball query grouping
and a hierarchical PointNet structure.

Several other works have defined a convolution operator for point clouds.
These models typically leverage voxelization or octrees. The former approach
converts point cloud data into a regular three-dimensional voxel structure,
facilitating standard convolution. The latter approach uses a tree structure
combined with varying sizes of voxels to capture different densities within
the point cloud. Notable examples of these models include SparseConvNet
(Graham et al., 2018) and OctNet (Riegler et al., 2017), which exploit the
natural sparsity of input point clouds. However, these techniques are limited
by the local receptive field of the convolution operator, which may be
insufficient for large-scale point clouds.

To address this limitation, Wang et al. (2019) introduced the EdgeConv
operator, which enables dynamic convolution over the point cloud and captures
both global and local information. Despite these advancements, convolutional
approaches still underperform compared to attention-based counterparts (Guo et
al., 2021), a topic we discuss in the following subsection.

###  2.2 Transformer Architectures

Transformer models, introduced by Vaswani et al. (2017), were initially
designed for natural language processing (NLP) tasks, exploiting the attention
mechanism. These models operate on tokens derived from the input through a
tokenization operation. The tokens are processed to obtain a hidden
representation, typically by a task-specific head, to yield the final
prediction.

The high parallelization possible in attention calculations has allowed
transformer models to benefit significantly from recent GPU performance
improvements, enabling larger models’ training (Narayanan et al., 2021).

Following their introduction in NLP, the general nature of their architecture
has allowed transformers to be employed across various fields, achieving
outstanding performance (Islam et al., 2024). Typically, the only adjustment
needed for a transformer to work with different data types is to replace the
initial tokenization module.

In image processing, for example, Vision Transformers (ViTs) have replaced
Convolutional Neural Networks as the state-of-the-art for various tasks, such
as image classification and image segmentation (Dosovitskiy et al., 2021;
Thisanke et al., 2023).

Graph Transformers (GTs) (Müller et al., 2023), a family of architectures
based on transformers, have been developed to operate on graph-structured
data, leveraging the desirable properties of the attention operation.

Transformer-based models have recently emerged in point cloud data processing
(Zeng et al., 2022). In particular, Guo et al. (2021) and Zhao et al. (2021)
simultaneously introduced a transformer-based model for point cloud data.
Transformers inherently possess desirable properties for point cloud deep
learning, as they naturally maintain permutation invariance among the tokens,
treating them as an unordered set.

Due to the inherently quadratic nature of the attention operation (Duman Keles
et al., 2023), scaling the number of tokens used in transformers is
challenging. One solution involves developing dynamic models capable of
subsampling only the most relevant tokens in the input.

In the field of image processing, models such as AdaViT (Meng et al., 2022)
and DynamicViT (Rao et al., 2021) have been developed, proposing methods for
selecting and removing unimportant tokens during the calculations.

In the context of point cloud data, recent works, such as that by Yang et al.
(2019), employ strategies analogous to those used in ViTs to subsample less
relevant tokens.

Existing models have successfully enhanced the performance and
interpretability of their base models, albeit at the cost of some accuracy.
These models, applicable to images and point clouds, necessitate the pre-
determination of a computational budget, establishing a fixed trade-off at
inference time. The contribution we introduce in the subsequent section offers
a significant advancement over these models. Our approach allows for an
adaptive computational budget at inference time, enabling a dynamic trade-off
between accuracy and computational complexity. Importantly, this adaptability
is achieved without the need to re-train the model.

##  3 Adaptive Point Transformer

In this work, we present the Adaptive Point Transformer (AdaPT), a model
capable of classifying point cloud data using a flexible computational budget
at inference time. To achieve this, we pair a series of transformer layers
with a set of token selection modules. These modules subsample, in a
differentiable manner, the tokens used in the point cloud transformer,
reducing the computational effort needed in subsequent layers.

Figure 1: Architecture of the AdaPT model. The embedding module consists of an
Absolute-Relative positional embedding (Yang et al., 2019). The transformer
blocks are paired with the proposed drop predictors. Finally, the
representation is processed by a classifier head which outputs a prediction.

The amount of tokens sampled at each layer is not fixed. This allows for the
computational budget available to the model to be decided at inference time
for each data point. In the following sections, we explain the components of
the model, which are shown schematically in Fig. 1.

###  3.1 Point Cloud Transformer

We want to build a transformer-based architecture in order to be able to drop
less relevant tokens in a similar manner to Rao et al. (2021). We need our
point cloud data to be embedded in tokens to employ a transformer model. We do
this through the Absolute-Relative Positional Embedding (ARPE) module
presented in Yang et al. (2019). This module embeds each point into a token
utilizing information from the point itself and its nearest neighbors. Let
$x\in\mathbb{R}^{N\times F}$ be a point cloud made of $N$ points, each with
$f$ features. Let $k_{i}\in\mathbb{R}^{k\times F}$ be the set of $k$ nearest
points to a point $x_{i}\in x$. The embedding is done as follows:

$\text{ARPE}(x_{i})=\gamma(\max\\{h(x^{\prime})\mid x^{\prime}\in k_{i}\\})$

where both $\gamma$ and $h$ are Multi-Layer Perceptrons (MLPs) with group
normalization and ELU activation (Clevert et al., 2016).

The tokens are then processed by a series of transformer layers employing
multi-head attention (MHA), layer normalization (LN), and a MLP composed of
two linear layers with GELU activations:

$\displaystyle z_{0}=\text{ARPE}(x_{i})$ $\displaystyle
z_{\ell}=\text{MLP(LN(MHA(LN}(z_{\ell-1}))))$

The final token representations are averaged, and the result is passed through
a classification head made of a MLP composed of two linear layers with GELU
activations, which outputs the network’s prediction:

$z^{\prime}_{f}=\frac{1}{n}\sum_{i=0}^{n}z_{f,i}\qquad\qquad
y=\text{MLP}(z^{\prime}_{f})$

where $z^{\prime}_{f}$ is the average computed over the final tokens.

###  3.2 Drop predictor module

The main aim of the AdaPT model is to reduce the computational effort during
inference by subsampling the tokens. This is done through modules responsible
for the token selection, which are called “drop predictors”, whose
architecture is shown in Fig. 3. These modules take into consideration both
local (relative to the single token) and global (relative to all tokens)
features. Let $x\in\mathbb{R}^{n\times F}$ be the input tokens to a drop
predictor layer. We first compute what we call local and global features as:

$z_{local}=\text{MLP}(x)\in\mathbb{R}^{n\times F^{\prime}}\qquad
z_{global}=\text{Agg(MLP}(x))\in\mathbb{R}^{F^{\prime}}$

where Agg is an aggregation operation, e.g. average pooling, and $F^{\prime}$
is an arbitrary dimension, in our case, $F^{\prime}=F/2$. The global features
are repeated $n$ times, obtaining a new $z_{global}\in\mathbb{R}^{n\times
f\prime}$. These features are concatenated and fed to an additional MLP that
predicts the probabilities of keeping the tokens:

$z=[z_{local},z_{global}]\in\mathbb{R}^{n\times
F}\qquad\qquad\pi=\text{Softmax}(\text{MLP}(z))\in\mathbb{R}^{n\times 2}$

where $\pi_{i,0}$ denotes the probability of eliminating the $i$-th token,
while $\pi_{i,1}$ denotes the probability of keeping it.

Figure 2: Scheme of the drop predictor module architecture.

Figure 3: Drop predictor modules usage in the PCT.

We also want the model to learn to select the least relevant tokens for a
given task and eliminate them. For end-to-end training to be possible, we need
to sample the tokens in a differentiable manner. This is done by employing the
Gumbel-Softmax (GS) estimator (Jang et al., 2017) to differentiably sample
from the probability distributions $\pi_{j}\in\mathbb{R}^{2};j=1,...,N$:

$\text{GS}(\pi_{j},\tau)_{i}=\frac{\exp((\log(\pi_{j,i})+g_{j,i})/\tau)}{\sum_{k=0,1}\exp((\log(\pi_{j,k})+g_{j,k})/\tau)}\quad
i=0,1;j=0,...,N$ (1)

where $g_{i,j}$ are i.i.d. samples drawn from Gumbel(0,1)111The Gumbel
distribution samples can be drawn by computing $g=-\log(-\log(u))$, where $u$
is drawn from Uniform(0,1)., and $\tau$ is the temperature parameter, which is
set to $\tau=1$ in our experiments.

To obtain a binary output an argmax is applied during the forward pass, while
straight-through estimation (Jang et al., 2017) is used for the computation of
the gradients. We obtain the binary mask $D\in\\{0,1\\}^{N}$ we use to index
the tokens as:

$D_{j}=\text{argmax}(\text{GS}(\pi_{j},\tau))_{1}\in\\{0,1\\}\qquad j=1,...,N$
(2)

We use only the values corresponding to keeping the tokens so that $D$ can be
used directly as a boolean mask to index the tokens. We add drop predictor
modules to a subset of the transformer blocks in the PCT, as shown in Fig. 3.

During the training phase, the tokens are not eliminated; they are instead
masked in the attention, effectively preventing them from contributing to the
prediction in successive steps. At inference time, the tokens are completely
removed, boosting the computational efficiency in the successive layers. A
demonstration of the effect of this token elimination is shown in Fig. 4. Each
drop predictor module also passes its decision on which tokens to eliminate to
the next layers in such a way as to ensure that, during the training phase,
once a token is masked, it stays masked for all of the remaining computations.

Figure 4: Visualization of the kept tokens along the model’s layers for a few
representative examples.

Additionally, we add a regularization term to the loss to incentivize the
model to drop tokens. We describe the regularization term for a single budget
here, and its extension to a flexible budget in the next section. Denote by
$t_{i}$ the target ratio of dropped tokens for the $i$-th drop predictor. The
regularization term we consider is the following:

$\mathcal{L}_{drop}=\sum^{p}_{i=0}\frac{(d_{i}-t_{i})^{2}}{p}$ (3)

where $p$ is the total number of drop predictors, while $d_{i}$ and $t_{i}$
are the fraction of token dropped and the target drop ratio for the $i$-th
drop predictor.

This regularization term is added to a standard cross entropy loss term
$\mathcal{L}_{CE}$. The complete loss function is then:

$\mathcal{L}=\mathcal{L}_{CE}+\alpha\mathcal{L}_{drop}$ (4)

where $\alpha$ is a parameter controlling the strength of the regularization.

###  3.3 Flexible budget

Finally, we want our trained model to be able to operate with multiple
computational budgets, which are to be selected by the user at inference time.
To achieve this, we add to the model multiple sets of drop predictors in
parallel, one for each desired budget. Note that each drop predictor layer (as
described in Section 3.2) is extremely small, thus adding a low overhead of
parameters to the complete model. These $B$ sets of drop predictors are all
trained simultaneously while sharing the same backbone model. This is shown
schematically in Fig. 5.

Figure 5: The budget parameter selects the set of drop predictors to be used
in the classification of a point cloud. This parameter also selects a specific
regularization term that is used to train the corresponding drop predictor
set.

Let $b\in\\{0,...,B\\}$ be an index variable denoting increasingly higher
budget values set by the user. We use $t^{b}_{i}$ to denote the corresponding
$b$-th desired budget at the $i$-th layer. In addition, denote by $d^{b}_{i}$
the ratio of dropped tokens at layer $i$ if we use the $b$-th drop predictor
module. The regularization term in (3) becomes:

$\mathcal{L}_{drop}=\sum_{b=1}^{B}\sum^{p}_{i=0}\frac{(d_{i}^{b}-t^{b}_{i})^{2}}{p}$
(5)

where we want all budgets to be simultaneously satisfied depending on the
index $b$. Computing this regularization term is unfeasible because it would
require $B$ forward passes for each input point cloud (one for each budget).
To this end, during training a random budget is extracted for each batch of
data, and the corresponding regularization term and set of drop predictor
modules are used for that batch.

During inference, the budget is an additional input required by the model,
which selects the appropriate set of drop predictor modules. By selecting a
set of modules trained with a certain regularization term, this parameter
selects the ratio of tokens that will be eliminated during inference, hence
selecting the computational budget for that prediction.

##  4 Experimental section

In this section, we compare our model to two other transformer-based
architectures for point cloud classification, presented in Guo et al. (2021)
and Zhao et al. (2021). The dataset used for these comparisons is ModelNet40
(Wu et al., 2015). To allow for a fair comparison of the results, we
implemented the models used in these works into our pipeline, training and
testing the models on the same split of the dataset and with the same data
preprocessing.

###  4.1 Regularization target and Budget parameter

We set the target used for the regularization presented in (3) using two
hyperparameters: the number of layers with a drop predictor associated,
$\ell$, and the desired ratio of dropped tokens to initial tokens after the
last layer, $\rho$. Once those parameters are fixed, the target ratio for each
layer is computed as:

$t_{i}=\frac{i}{\ell}\cdot\rho\qquad i=1,...,\ell$ (6)

Additionally, we need to set the target for different values of the budget
parameter. We choose to have the target depend linearly on the value of the
budget parameter, so that the only hyperparameter to be set is the number of
different budgets available, $B$. The targets associated to the different
budgets are then computed as follows:

$t_{i}^{b}=\frac{B-b}{B-1}\cdot t_{i}=\frac{(B-b)\cdot
i}{(B-1)\cdot\ell}\cdot\rho\qquad b=1,...,B\quad i=1,...,\ell$ (7)

We set $\ell=4$, $\rho=0.8$, and $B=4$ in our experiments. With our parameters
choice, we obtain the targets in Tab. 1.

b i | 1 | 2 | 3 | 4
---|---|---|---|---
4 | 0 | 0 | 0 | 0
3 | 0.07 | 0.13 | 0.20 | 0.27
2 | 0.13 | 0.27 | 0.40 | 0.53
1 | 0.20 | 0.40 | 0.60 | 0.80
Table 1: Values of $t^{b}_{i}$ for the hyperparameters chosen in our
experiments. Every row corresponds to a set of drop predictors associated with
the value $b$ of the budget parameter.

Each row of Tab. 1 corresponds to a target that will be used to train a set of
drop predictor modules.

###  4.2 Point Cloud Classification on ModelNet40

ModelNet40 consists of 12,311 CAD models distributed across 40 object
categories, making it a popular choice for benchmarking point cloud shape
classification. To ensure a fair comparison, we adopted the official split,
allocating 9,843 objects for training and 2,468 for evaluation. We adopted the
same sampling strategy described by Charles et al. (2017) to uniformly sample
each object to 2048 points.

During training, a random translation in the range $[-0.2,0.2]$, a scaling in
the range $[0.67,1.5]$, and a random input dropout were applied to augment the
input data. During inference, no augmentation is applied to the data.

A batch size of 64 was used during training. The initial learning rate was set
to 0.001, with a cosine annealing scheduler adjusting it at every epoch.

The experimental results for this classification task are presented in Table
2. We can see that our model performs similarly to analogous transformer-based
models. More importantly, we can see that our model does not suffer a
significant loss of performance when the token drop is introduced. We can also
see that the performance is approximatively stable across all the different
tested budgets, as shown visually in Fig. 6. We underline again that the
results in Fig. 6 with the same model while varying the budget as an input
parameter.

Model | Accuracy
---|---
PCT | 91.0
Point Transformer | 90.8
$\text{AdaPT}_{b=4}$ | 90.1
$\text{AdaPT}_{b=3}$ | 89.8
$\text{AdaPT}_{b=2}$ | 89.5
$\text{AdaPT}_{b=1}$ | 89.4
Table 2: Shape classification results on the ModelNet40 dataset. Figure 6:
Model performance when selecting different budgets. A stronger budget means
more eliminated tokens at each layer. We can see that the model manages to
keep a stable performance when the intensity of the budget varies.

###  4.3 Flops count evaluation

We evaluated the inference FLOPS required by our model while varying the
number of initial tokens.222To evaluate the computational efficiency of our
model, we calculate the inference flops using the FlopCountAnalysis method
from the fvcore library. Even though the regularization term in (3) pushes the
model towards the desired number of dropped tokens, this number is not fixed
during training. During inference, the number of tokens to be kept at each
layer is deterministic, and the dropped tokens are completely eliminated,
making it possible to count the model’s flop reliably.

Figure 7: Flops in function of the number of tokens in the transformer part of
the model.

The results are shown in Figure 7. As expected, our model is significantly
more efficient when eliminating tokens.

We further investigated the effect of token selection on the model’s
performance. We trained the model with different values of the hyperparameter
$\rho$ and observed the accuracy on the ModelNet40 classification task. We
only consider the highest possible value for the budget parameter in these
experiments. The result of these experiments are shown in Figure 8. As
expected, we observe a trade-off between model accuracy and number of dropped
tokens. We also observe a significant drop in performance when the drop target
is set too high, requiring the model to eliminate $95\%$ of the tokens in the
last layer.

Figure 8: Accuracy of the model in the shape classification task on the
ModelNet40 dataset as a function of the $\rho$ hyper-parameter.

###  4.4 Ablation studies

In this section, we present additional experiments aimed at investigating the
effectiveness and robustness of our model. To conduct these experiments, we
fix the budget parameter to $b=1$, corresponding to the most severe drop
targets.

To test the effectiveness of our sampling strategy, we replace it with
alternative, simpler sampling methods. In particular, we tested two different
setups as alternatives to our method of token selection:

  * 1.

Random Selection: the drop predictor selections are replaced by random
samplings of the tokens. This can be seen as a form of token dropout.

  * 2.

Farthest Point Sampling: We perform farthest point sampling on the tokens, in
place of the selection made by the drop predictors. We perform this sampling
by computing distances with respect to the original points’ tridimensional
positions.

Both performed worse than our sampling strategy. This indicates that our
sampling strategy succeeds in selecting more informative tokens than the other
two non-learnable sampling methods. The results of this test are shown in
Table 3.

Sampling strategy | Accuracy
---|---
Random | 79.5
Farthest-Point | 85.2
Adaptive | 89.4
Table 3: Shape classification results on the ModelNet40 dataset using
different sampling methods.

##  5 Conclusions

In this paper, we presented Adaptive Point cloud Transformer (AdaPT), a model
for efficient point cloud classification. We employed an end-to-end learnable
and task-agnostic sampling mechanism relying on the Gumbel-Softmax
distribution to achieve differentiable sampling. Moreover, we designed a
budget mechanism capable of regulating the model’s inference FLOPS without
retraining. Results on the benchmark dataset ModelNet40 for a point cloud
classification task demonstrate the effectivness of the proposed model.

Future developments of this work include extending the model’s task capability
to include point cloud segmentation and further investigating the impact of
learnable point subsampling on the model’s interpretability.

## Declaration of competing interest

No author associated with this paper has any potential or pertinent conflicts
which may be perceived to have impending conflicts with this work.

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