SAND: Smooth imputation of sparse and noisy functional data with Transformer networks
Authors: Ju-Sheng Hong
Junwen Yao
Jonas Mueller
Jane-Ling Wang
Venue: NeurIPS 2024 poster (2024)
Keywords: functional data
sparse functional data
imputation
Transformer
self-attention
Primary area: machine_learning_for_other_sciences_and_fields
TL;DR: We introduce a new self-attention module (SAND) to impute sparse functional data in a differentiable manner.

Abstract: Although the transformer architecture has come to dominate other models for text and image data, its application to irregularly-spaced longitudinal data has been limited. We introduce a variant of the transformer that enables it to more smoothly impute such functional data. We augment the vanilla transformer with a simple module we call SAND (self-attention on derivatives), which naturally encourages smoothness by modeling the sub-derivative of the imputed curve. On the theoretical front, we prove the number of hidden nodes required by a network with SAND to achieve an $\epsilon$ prediction error bound for functional imputation. Extensive experiments over various types of functional data demonstrate that transformers with SAND produce better imputations than both their standard counterparts as well as transformers augmented with alternative approaches to encode the inductive bias of smoothness. SAND also outperforms standard statistical methods for functional imputation like kernel smoothing and PACE.

SAND: Smooth Imputation Of Sparse And Noisy Functional Data With Transformer Networks
Ju-Sheng Hong"
Junwen Yao Department of Statistics University of California Davis Davis, CA juyao@ucdavis.edd
Department of Statistics University of California Davis
Davis, CA 95616
jsdhongducdavis..cm
Jonas Mueller Cleanlab Cambridge, MA jonaswmelller@maal.cco com
Jane-Ling Wang
Department of Statistics University of California Davis Davis, CA j janelwang@ucdavis edu
Abstract
Although the transformer architecture has come to dominate other models for text and image data, application to irregularly-ppaced longitudinal data has been limited. We introduce a variant of the transformer that enables it to more smoothly impute such functional data. We augment the vanilla transformer with simple module we call SAND (self-attention on derivatives), which naturally encourages smoothness by modeling the sub-derivative of the imputed curve. On the theoretical front, we prove the number of hidden nodes required by a network with SAND to achieve an prediction error bound for functional imputation. Extensive exper- iments over various types of functional data demonstrate that transformers with SAND produce better imputations than both their standard counterparts as well as transformers augmented with alternative approaches to encode the inductive bias smoothness. SAND also outperforms standard statistical methods for functional imputation like kernel smoothing and PACE.
Introduction
Functional data analysis (FDA) offers framework for analyzing complex data sampled from random functions or curves. Such data are encountered in many applications including air pollution studies, fMRI scans, growth curves, and sensor data from wearable devices. FDA has thus grown into established field in statistics [14] 9 24]. Functional data intrinsically have infinite dimensions but tractably handled by assuming they were generated from smooth underlying process. The simplest form is a univariate random curve x(f) defined on compact set over the real line R. Without loss of generality, it is typically assumed that x(t) is smooth stochastic process over the [0, 1] interval domain. Let 2; (t), E {1..... n be random sample of xt)), for instance, growth curves corresponding different subjects discretely over the time interval [0, 1], and often only at few irregularly spaced time points, namely (t: 14 Moreover, the observed measurements are often noisy. Thus, the observed data are [[(1)]) Miy ) in where Yij = x,(1,g) + 6y and 6jj are random noise effects. For many
population. In practice, each (t))) can only be observed
Conrespondence: Ju- Shenhhn Hong @jddinngidgongnnucaaaaee Code available at: httpt::// http:///gtttbb..t com/ wctthhab.cmm/sshhoogo////aa
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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applications, it is reasonable to assume that the noise is independent over t and across subjects and follows zero mean distribution with finite variance.
The objective of this paper is to recover the latent processes rz((), E {1.....n]] using the noisy/sparse observed data D. We introduce estimator that is effective for both finely and sparsely measured functional data (i.e. regardless of whether the number of measurements for the i-th subject, T4, tends to infinity or is uniformly bounded by constant). The latter case, sup < oc, is common in longitudinal studies and called sparse functional data in FDA. Accurate recovery (imputation) of the underlying function from noisy/sparse observations can improve analyses of longitudinal data across many sciences.
The conventional approach to imputing longitudinal data is employ parametric model with mixed- effects, such as the linear mixed-effects model. This approach lacks the flexibility to capture the shapes of more complex trajectories, data. For instance, MICE [21] uses a chained equation approach, where the imputation of each missing value is conditioned on the imputed values of the other variables, resulting in more accurate and stable imputations compared to single imputation methods. However, the resulting imputed functions may not be smooth and to get stable imputations, many iterations are required if the percentage of missing values is high 17 25].
nonparametric method is preferred for imputing functional
A standard FDA approach that avoids this drawback is to first reduce each function to a finite but possibly growing dimensional vector and then reconstruct the trajectory of each subject using a particular set of basis functions associated with the dimension reduction approach (28). Two widely used dimension reduction methods are the basis expansion approach [16 3 and Functional Principal
Component Analysis (FPCA) 2 15
T8 10 27
12 26 Among these, FPCA provides the most
parsimonious approach. The most popular FPCA approach is PACE 24 4 which estimates the top principal component functions and the corresponding scores of each subject reconstruct the latent process of this subject through the Karhunen-Loève decomposition.
All aforementioned approaches use the combined data from all subjects to conduct the imputation for each particular individual (-). Another approach that works well for densely observed functional data is to apply smoothing method to interpolate the data from individual. While such smoothing does not borrow information across subjects, it can be effective when the numher of measurements (n,) is large for all subjects (proper smoothing can denoise the observations).
Recent advancements in deep learning for functional data imputation include techniques like neural processes [ and GAIN [29). Their applications to real data are however dwarfed by the widespread use of transformer networks [23]. Using a self-attention mechanism to discerningly attend to regional information across a domain, transformers offer natural architecture for learning the right information to rely on in order to best estimate (t) at t. Specifically, encoder-decoder transformers representations generated by the encoder to produce properly contextualized estimations through the decoder. We note that imputing sparsely observed functional data can be viewed as a sequence- to-sequence learning task, for which transformers should be well-suited. However, imputation via transformers fails to yield a naturally smooth function. In this paper, we introduce a small modification of transformers that addresses this issue and produces more accurate imputations.
Summary of our contributions. (I) We show empirically that a vanilla transformer is promising for functional data imputation- it outperforms PACE and neural processes but the imputations are not differentiablessmoothh (2) We introduce novel network layer. SAND (self-attention on derivatives), that can be learned end-to with the rest of transformer architecture. Extending the attention module, our SAND layer guarantees smooth imputations and does not require any assumptions on the data distribution. (3) We prove that imputations from SAND are first-order differentiable (Theorem D. and derive an analytical expression relating the number of hidden nodes and prediction error of a SAND network fit via empirical risk minimization (Theorem
2 Related works
2.1 Sparse and intensively measured functional data
In most longitudinal studies involving human subjects, the number of measurements n; is often small (bounded). This type of data is called sparse functional data. When n; is large and grows with the sample size to infinity, such functional data are called intensively measured functional data. We
2

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note that they are not time series data, which is collection of data observed discretely on an equally spaced time grid for single subject. There is no smooth latent process that generates this time series.
For intensively measured functional data, we can apply a kernel smoother to smooth the noisy observations individually for each subject recover the underlying curve x; (t) and perform subse- quent data analysis. Let (-) be kernel function (often a symmetric p.d.f). (;(() is estimated by the weighted sum of observed data,
) where weights are computed using kernel K (-): (1) Σド= (4:) (4yy with (+)+) = K(//h//h. The bandwidth that determines the size of the related neighborhood can be decided by generalized cross-validatte
31
It is not possible to consistently reconstruct the underlying curves for sparse functional data, due to the limited number of available observations per subject. It is however possible consistently estimate the mean (u()) = [[x(()] and covariance function G(s, () = Cov(x(s), x())) from sparse functional data [27]. Once we have a consistent estimate of the covariance function, we can perform functional PCA and get the imputation of z, (t) as done in the popular PACE method (27). Letting
{(A,, (x{-))]==1 denote the eigen-com ceeeonnnssssts of the kernel operator defined by G(s, t) (which can
all be estimated consistently), PACE uses the Karhunen- Loéve series expansion of individual curves,
r1 = (())=
Σ (
and replaces all unknowns with their estimates. To estimate the principal component scores Eik. PACE uses the best linear predictor A E 14), where du (ox((ti))) y: = (w1))]=1
When (=2(-) is Gaussian process, we expect the best linear predictor PACE to shine and this is reflected in the empirical results in Tables SS and [STO] These results further reveal that PACE continues to outperform existing imputation methods, such as MICE 22] CNP 7 and GAIN 29). when (-) is not Gaussian. This outstanding performance underscores the status quo, that PACE is the state-of-the-a imputation method for functional data. However, PACE is linear imputation method, which may restrict its ability to estimate a more complex data structure. In this work, we aim to replace this constraint with more flexible model that uses our custom self-attention module.
2.2 Neural processes
The first study of deep learning for functional data imputation led to neural processes (NP) 777. which perform imputation by modeling the distribution of a target x; (t) conditioning on the observed contexts {(ty }ir] via latent variables. From a set of observed data, the NP estimates distribution over functions from which these data points could have been sampled. This estimation is done more flexibly than traditional Gaussian Process inference by utilizing a learned neural network. Variants of this approach include the conditional neural process (CNP) [7] and attentive neural process [II].
While the NP appears promising, empirically find that straightforward modeling via transformers yields more accurate functional imputations. Our proposed transformer extension learns to recover smooth trajectories from fixed set of sparse observations in a single-shot. NP models are more complex and must learn their conditional distributions by iteratively sampling observations from the process. At each training iteration of an NP, one first samples a random function and then some observations from the function, repeating this procedure many times. In contrast, our method specifically aims recover smooth trajectories from given (fixed) set of observations, rather than relying on iterative sampling.
2.3 Sell-attention
Transformer 23 comprises an encoder and a decoder, each consisting of stacks of attention and feed-forward layers. (30 highlight the roles of these layers: self-attention captures contextual mappings of the inputs and the feed-forward layer maps them to the desired output values. To provide a general overview of the self-attention module, we consider a generic real-valued input matrix X of size q m. Let ha be the number of hidden nodes and H be the number of heads used in self- attention module. For each head h, parameters (((h) E Roxhaa and the rey-value-query component (h)
(h) ghaxy are learnable. Denote softmax(X) as applying the softmax operation (k) w(( w(fh)
3

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to columns of X. A self-attention module defines a mapping from Rixm to R
H
Attn(X)= = X + Σ A=1
whth)
( softmax
(1) V X
w (h) K
X
x
/ Vha
(I)
The input X can be viewed as 2 x n; matrix with the first and the second rows being (Miy)}= and (t.y (t respectively. The softmax operation generates a matrix with non-negative entries, where the columns sum up to 1. By multiplying softmax on the right of X, we obtain a weighted sum of the columns of X. This can be viewed as an interpolation method aiming to recover the underlying process (t) simultaneously for all subjects, by pooling all observed data together estimate the weights Wx and Wo. This approach proves to be more efficient and distinct from individually interpolating each subject through a weighted sum of X.
3 Method
As our objective is to smoothly impute curves, transformer with ReLU activation function (RT)
may not be the optimal choice as ReLU not differentiable O. An alternative is to consider GeLu-activvted transformer (GT). Since GT outperforms RT in our benchmarks (Tables SS2 and SS3.
we investigate GT and its variations and propose an architecture that equips GT with our SAND
module in the rest of this section.
3.1 Functional imputation via the vanilla transformer network
Let I = {0 M-1 M-1 :- 1} be a discretization of [0, 1] where M is the number of points, T; be an imputation the cardinality. A transformer network defines a mapping from R2x== to R through T, - (S.) where S₁ (source) the input and T, (target) is the output. During training, the mean squared error (MSE) loss between the observations y, and the imputations on }= is minimized. Let ||0|| denote the & norm of æ. We use the same T, as the imputation projected on (tyy)²= and minimize
output grid I for subject and A be the training index set with ||Allo being
MSEA = >> IT. |:ll//|A||oo |Al|oo-
EEA
During the imputation, we select a sufficiently large M 101 to ensure that T' accurately represents the imputation within the interval [0, 1]. However, due to the universal approximation capabilities of transformers (30] and the presence of noisy data during training, well-trained models can often produce zigzag-like like imputations that mimic the noise in the training data. Meanwhile, it's essential to note that when these models encounter testing data, which is also observed with noise, they may continue to generate zigzag patterns if they have learned to replicate the noise patterns present in their training data, as shown in the first two plots of Figure [ This can result imputations that fail to capture the underlying, noise-free structure of the data.
A natural way to counter non-smooth issue is to add penalty. Let 3 denote the collection of all parameters used in GT. By minimizing MSE, + ||] where is chosen by the validation dataset, we reduce the complexity of the model by shrinking the parameters towards zero. This results in a model that is less likely to fit the noise. However, it does not guarantee the smoothness of the target function (see the third plot of Figure D Alternatively, we explicitly penalize the output function by considering the objective function as MSEA + A EiEA, S'' (T(2 (0)² di/ |l This objective function penalizes the curvature of the imputation and restricts its behavior. However, as illustrated in the fourth plot of Figure [. this penalization approach fails to resolve the non-smooth issue.
Another approach is to apply statistical methods that ensure the differentiability of the imputations from transformers (either penalized or not). For instance, after the training of the network, we may apply PACE to estimate the eigen- components of the non-smooth outputs and use them to reconstruct the prediction output. A kernel smoothing can also individually smooth each of the outputs. However, transformers with these methods Related numerical results can be found in Section 5
typically preferred due to their lack of end-to-end training.
In Porch,, we torch.diff and cumsuun efficiently calculate the derivative and the integral.

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GT+2N
GT-120
SAND
Figure 1: Close-ups of imputations generated by transformer variants (red curves) for testing data. Green lines are the true underlying functions from which observations (blue dots) are sampled with measurement errors. +T+ and +T+ are GTss regularized by the & norm of parameters and the curvature of imputations, respectively. SAND is our proposed method.
Remark 1. Here, incorporate positional encoding for functional data and prepare it as input sequences for a transformer. Let M'ax max, n4, For each subject i, the locations 4; (4g))²- and values y; = (1t5))=1 are both padded with (Amax n2)) zeros. To prevent attention modules from attending to the padding. we use mask vectors the encoder and decoder components. To incorporate positional encoding, we define E; E R as the encoding of t, with being the encoding dimension. (E,) is computed as sin ((M - 1) htt//10000/+///// for odd values of j.
and cos ((M 1) 100000//))/ for even values of j. Finally, stack S [y t; E;] input.
3.2 Self-attention derivative
To overcome the lack of smoothness in imputations, we introduce a new module called Self-AtteNtion on Derivative (SAND). SAND is stacked after GT, taking the imputations from GT as inputs, and outputting a smoother imputation in two steps: it first learns the derivative of the original imputation through novel operator Diff(-), which fathoms the derivative information and is inspired by the self-attention module in I then, reconstructs a smooth version the imputation via the numerical integral operator Intg(-). We describe these two operators after introducing necessary notations.
Let 7 be imputation grid I with the first element (7); and T. T (T); T being shifted by (T) Subsequently, T and respectively, and the rest of rows are the p-dim positional encoding of I. The symbol cumsum(rr (x) is the vector of the cumulative summation of æ. Its j-th element [eumsum(2:)], is Σk Zg. Let W(())
(1 + p)-by-M matrices where the first rows are T and To.
w (b) w (h) and w be learnable parameters. The Diff and Intg operators are:
DT D Σ w (w("
[ (
(w!h T.)
/VTaa
A=1
Intg(D) cumsum| [D/(M - 1)].
(4)
Denote T as the output from SAND. Then, SAND can be precisely summarized as follows:
T SAND(T) (T), + Intg]Diff(T)).
55)
Figure 2 depicts the role of SAND: takes an output T from transformer, learss its derivative via Diff and Intg, and recovers the noise-free trajectory by minimizing the & distance of and T:.
Mmm
Vanilla Transformer
Diff
Intg
SAND
SAND's pipeline. Dashed lines are underlying processes, dots are observations sampled with errors, solid red curves are imputations, and solid orange curves are learned derivatives.
Figure 2:
(-ff(-) operator. The structure of Diff(-) is similar to the attention module defined in [ where the key-query product scaled by Vha to prevent the product from growing with the hidden dimension. However, Diff(-) removes the softmax operator the key-query product because the product is used as weights while approximating derivatives. It should not be restricted to be positive only as for any differentiable f and provided that b is close to
the approximation of fa( ~
5

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involves a weighted sum of f(a) and f(b) with both positive and negative weights. We remark that despite we don't 't have the observed derivative information to guide Diff(-) during the training process, we conduct a comparison using simulated data. Figure SST compares the output from Diff with the derivative of the underlying process and indicates that Diff(.) adeptly captures the overall shape.
Intg()) operator. The function Intg()) approximates the integral f(b) f(a) = S." fz(x) dx, where f() is differentiable. At a high level, because Diff(.) is a continuous operator and the integration of a continuous function is differentiable, the output of Intg(-) is continuously differentiable. This result is formally developed in Theorem [ Furthermore, the Intg(-) operator allows SAND to impute the target process from any index ko- We let the koth location serve the purpose of (a, (())) in the integral at the beginning of this paragraph, then the imputation from SAND at index ki is Tho + (-1) [(k <ka) EkEx: (xex:(D)</( (D) 1) where I [A: < ko) is if 11 < ko and O otherwise, and IC is the set of indices between kn and k1 (inclusive). Hence, SAND imputes trajectories in bidirectional fashion. A similar approach, scheduled sampling, was explored by [I]
Remark 2. Conceptually, any machine learning model that handles vector inputs and outputs could be utilized to enhance the coarse imputation from GT, as there are numerous viable options. The attention module is adapted for two reasons:
Computational Efficiency. SAND is computationally efficient compared to other well-established vector-to vector models like recurrent networks or convolutional networks. This efficiency is crucial in our choice, as highlighted by 23 The attention module itself has demonstrated effectiveness across diverse applications, thanks in great part to its scalability.
Achieving Smooth Outputs. Our objective is produce a smooth curve, which requires that its first derivative be continuous. Since most neural networks inherently model continuous functions, they are suited to this task. The first derivative (a) RJ f(b)-((a) involves linear combination of function values. In constructing SAND, we chose the attention module because it inherently performs
a weighted summation of its inputs, aligning well with our needs for modeling derivative.
4 Theoretical analysis
Theorem demonstrates that the imputations from SAND are continuously differentiable. Theorem 2 establishes a crucial finding that highlights the improved accuracy of SAND: it can be viewed as dimension reduction technique. Proofs of all results in the section are provided in the supplement. To begin Theorem 1 let C' (I) denote the set of continuously differentiable functions defined on I.
Theorem 1. Let T denote the outputs from SAND. Then, f is in C' ([0,, 1]) when M 00.
Theorem @ Timplies that the imputation from SAND is continuously differentiable. This is in contrast to GT, which is prone to overfitting and non -onssmotthsess issues, as discussed in Section 3.1 The assumption that M - do is fairly mild as it could tend to at any rate, not depending size n. In simulations, we validate our choice of M by reporting the total variation (TV) of the difference between an imputation and its underlying process. This measure is defined as the average of absolute differences between adjacent elements in the difference.
the sample
Since SAND aims at minimizing the l2 distance between its output (7) and the imputation derived from GT (its input, T), it is possible to drive its training error to zero by using a large number of hidden nodes (h,). However, this needs to be avoided as the input T contains random noise. Instead, the goal for SAND should be to effectively disentangle the inherent smooth process from the random noise in T. Therefore, a small training error c should be incorporated in SAND. The choice of is tied to the number of hidden nodes ha, which can be determined by the fraction of variance unexplained in FPCA (27). We formally encapsulate this intuition in Theorem
Theorem 2. Define A Tr, and Tre as in section 3 Suppose that { are the non-increasing eigenvalues of E, Tr.T.... For any 2 0. let de minfd / 2 1 c}. Then, there exists SAND satisfying: (1) ha 2 dei and (2) the positional encoding dimension being M, so that for any imputation output from SAND, we have
1
Immass 2 (T, TT)1B S
1T..el2
||Al|o
|
AEA
TEA

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where draak, the decoder masking for ith subject, is vector of length M whose jth element is 1 ift M-I is observed in subject i and is 0 otherwise and u & v represents the element-wise multiplication between vectors and u.
The eigenvalues of E, TT.TT.c e. SAND. As a consequence of Theorem 2 the following corollary shows the connection between SAND and dimension reduction techniques of FPCA.
Theorem are used to determine the number of hidden nodes in
Corollary 1. Let SAND be defined as in Theorem 2 with ha de. Suppose that for all subjects, we have full observations on the output grid I, i.e., all elements in dipuas is 1. Then, SAND minimizes
Σ EITT. T.,011 REA
IT
min Poran( (P) d. ||A.
|
iEA
Remark 3. Corollary | shows that SAND can seen as a dimension reduction technique by viewing ha and the fraction of variance unexplained and the number of principal components needed to attain an c-error in FPCA, respectively. However, because the presence of the decoder masking, ha and do not generally play the two roles. Still, Corollary D provides an intuitive insights of them.
5 Experiments
SAND is compared to seven baseline methods: PACE [27), FACE (2)., mFPCA [13, MICE [22).
CNP 7. GAIN (29. and IDS (one-dimensional kernel smoothing to impute each subject separately). PACE estimates the Karhunen-Loeww presentation based on the function input. FACE imputes the missing data through different covariance smoothing method from PACE. mFPCA uses a likelihood- based approach to estimate the functional principal components. MICE imputes the missing data based on the fully conditional specification, where each incomplete variable is imputed by a separate model. CNP parametrizes distribution over imputations giving a distributed representation of pairs of time points and functional values. GAIN imputes the missing data using generative adversarial nets. Due to the lack of available code, we did not include the methods of [16] and 10) We discuss more on them along with 133 in the supplementary materials.
Each baseline involves the choice of hyperparameters. In PACE and mFPCA, we let the fraction of variance explained 99%. FACE can select all hyperparameter automatically. MICE, we impute
20 versions of data and take their mean, as suggested in [25]. In CNP, we implement 7 [link] Jusing
PyTorch to train the model for 50,000 epochs. The code for GAIN provided [29 [link] We train the model for max of 10,000 epochs with 15 different random seed and take their mean. Finally, in IDS, the bandwidth of the smoothing window is selected by cross-validatio
All transformers in Section 3 were trained for 5,000 epochs using a mini-batch of size 256 with a learning rate of 10 and 15% dropout rate. The penalty parameters are picked from data-driven grid containing 8 points. In SAND, ha is set to 128 and the dropout rate is 5%. See the code in the supplement for details of the model configuration. To quantify the uncertainty of the imputations, we add the pinball loss with the 10th and 90th target quantiles to the MSE in (2) Training takes hours on one Nvidia 1100 with 10,000 samples.
Data splitting and evaluation of imputations. The dataset is divided into three parts: the first 90% is for training, the last 5% for testing, and the remaining 5% for validation. The error between prediction and the underlying process on the testing data is reported for comparison. To evaluate the performance of each method, we report the MSE testing data using 2 along with its standard error (SE). To validate Theorem I and our choice of M, we provide the mean and SE of the TV of the difference between any imputation T, and the underlying process (.).
5.1 Simulation studies
Through simulations, we show that transformer- based models outperform other imputation methods. Extensive simulation schemes varying the number of basis functions, the number of points sampled in a curve, the size of random noises, and the independence of the sampled points are considered. Our simulations are purposely designed to probe the shorts of all baseline methods and neural networks.
Data-generating process. Let (t) k [ask sin(2 + bu cos(2mktt) /k where t [0, 1], i e (1...., 104} and a,k. by follow zero-mean distribution. The actual observed data of the ith
7

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Table 1: MSE(SE) & TV(SE) on simulated data. Bold values indicate the top performing methods.
74
30
74 = 8 to 12
R; 3, 4,5
MSE(SE)
TV(SE)
MSE(SE)
TV(SE)
MSE(SE)
TV(SE)
PACE
189.9(4.3)
187.1(2.0)
450.0(15)
201.9(2.1)
795.5(33)
209.5(2.2)
FACE
284.6(8.8)
198.9(2.1)
488.2(16)
204.5(2.2)
807.1(32)
209.5(2.2)
mFPCA
224.7(5.8)
192.0(2.1)
480.3(16)
204.0(2.2)
787.1(31)
209.3(2.2)
MICE
176.7(3.7)
233.1(1.7)
721.6(27)
318.4(3.0)
1416(57)
332.7(2.8)
CNP
290.4(11)
198.9(2.0)
551.3(21)
207.6(2.1)
920.3(52)
211.9(2.2)
GAIN
261.9(6.8)
350.0(3.4)
1454(52)
413.1(5.1)
1862(51)
385.4(4.3)
IDS
262.9(6.0)
273.8(2.4)
735.3(22)
305.7(3.7)
1157(43)
263.3(3.1)
GeLU-activated transformers with penalties
GTI
169.8(3.2)
218.2(1.7)
436.7(15)
227.0(2.2)
798.6(35)
230.6(2.6)
GTIP
169.0(3.5)
179.9(2.0)
425.3(14)
199.4(2.1)
777.4(36)
210.2(2.2)
GTIS
169.8(3.2)
218.1(1.7)
436.7(15)
227.0(2.2)
796.5(35)
227.6(2.6)
GTIT
160.6(3.2)
171.9(1.)
421.8(14)
196.8(2.1)
783.4(34)
211.8(2.2)
GT2
174.83.55
223.0(1.7)
433.8(14)
221.9(2.1)
804.5(39)
226.1
GT2P
179.933.9
182.1(2.0)
422.6(13)
199,4(2.1)
788.9(39)
210.0(2.2)
GT2S
160.8(3.4)
168.5(1.5)
427.7(13)
208.8(2.1)
804.5(39)
12..4
GT2T
167.5(3.5)
173.3(1.9)
419.9(13)
197.1(2.1)
792.3(39)
211.0(2.2)
acttvactiv transformers with augmented modules
ATT
(35.1
220.0(1.7)
220.6(2.1) 446.9(14)
852.0(42)
224.0(2.5)
SAND
146.5(2.7)
164.6(1.8)
410.9(13)
196.8(2.0)
758.1(43)
206.8(2.2)
curve is a discrete data (ug)) where Vi = (tig) + E and Eig follows N(0, independently. The output grid [0, is discretized in [5/100] 100 in all simulation. The experiment consists of 12 scenarios, varying in the distribution of the eigenvalues (a.z and bıx), the number of basis functions 2K, and the magnitude of a². The full simulation is provided in the supplement (Table SI). In the main text, we only focus on the scenario when the cigenvalues follow an exponential distribution, which stimulates heavy-tailed processes, the number of basis functions set to 40, the time points within any subject are sampled independently, and the signal-to-noie ratio (SNR) is set to 4. For function input
f (t), we define ff as J f² dt/o" / Finally, each scenario comprises three cases determined by n2; either n, 30, n, = 8 to 12, or n. = 3, 4,5.
Transformer-based models. In Table 0 GTI and GT2 GT's penalized with & norm of parameters and the curvatures of the imputations, respectively, where the penalties are chosen by the validation dataset. For being or 2, GTTP, GTtS and GTiT use PACE, kernel smoothing and trend filtering [19]. respectively, to subsequently smooth the imputations obtained from GTi. GT variants are discussed in Scction 3.1 Moreover, investigate the impact of adding an extra self-attention module after the GT as it bears a similar structure to an interpolation method. We refer it as 'ATT' in our analysis.
Results. Table P the imputation MSEs and TVs (scaled by 1,000) along with their SEs for all models consistently perform the best across all benchmarks, indicating
4 cases. Transformer- their efficacy for functional data imputation. We begin by examining GTs, which empirically outperform other baseline methods (Table Despite this, imputations from penalized transformers still lack smoothness (Figure []. Table rreinforrss this observation by showing significantly higher TV in the imputations from GT1 and GT2 compared to imputations from PACF, FACF, and CNP.
This lack of smoothness can be further mitigated by subsequently applying methods like PACE or individual smoothing to further refine GT imputations. Table I confirms that total variations in all GTs decrease, yet closer look at the table reveals that PACE (GTIP and GT2P) can generally lower MSE, but falls short, particularly when n, is large, where PACE's posterior smoothing dramatically escalates the MSE. Individual smoothing (GTIS and GT2S) can improve imputations for dense functional data, yet has limited benefits for sparsely observed data. Although trend filtering methods (GTIT and GT2T) consistently reduce MSE and TV, their performance is ultimately surpassed by SAND. Lastly, using a self-attention module to boost GT doesn't yield positive results in our simulation. Figure D still displays non-smooth imputations, and MSES increase across all scenarios, compared to GT variants.
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Table 2: MSE(SE) and TV(SE) on real datasets. Bold values indicate the top 2 performing methods.
UK electricity
Framingham study
T2:
30
74:
12
14: =3,
3, 4, 5
Y1: to
MSE(SE)
TV(SE)
MSE(SE)
TV(SE)
MSE(SE)
TV(SE)
MSE*(SE)
TV*(SE)
PACE
12.8(1.8)
19.0(1.1)
30.1(4.5)
21.1(1.2)
39.6(5.2)
21.9(1.2)
0.076(0.00)
0.04(0.002)
FACE
15.8(2.1)
21.3(1.2)
32.5(5.4)
22.6(1.2)
39.6(5.2)
23.0(1.2)
0.075(0.004
0.03(0.002)
mFPCA
16.4(2.0)
22.2(1.2)
34.8(4.9)
23.2(1.2)
41.7(5.4)
23.3(1.2)
0.057(0.003
0.04(0.002)
MICE
20.4(2.2)
67.8(3.)
40.0(4.5)
65.4(2.8)
75.4(8.)
71.4(1.5)
0.199(0.011)
0.32(0.013)
CNP
23.0(3.5)
21.4(1.2)
31.5(4.3)
22.1(1.2)
47.9(7.1)
22.7(1.2)
0.087(0.004)
0.04(0.002)
GAIN
31.9(3.7)
108(5.6)
75.4(8.2)
104(6.7)
99.6(15)
121(2.4)
00100(),01)
0.23(0.007)
IDS
17.3(22)
19,4(1.1)
50.007.0)
22.8(1.3)
105(18)
44.1(2.7)
0.075(0.004)
0.05(0.003)
GT 10.7(L.8)
20.6(1.1)
31.2((.3)
23.2(1.3)
42.6(5.6)
38.5(2.5)
0.004(0.0002)
0.13(0.005)
SAND
10.0(1.9)
15.7(0.9)
26.7(3.0)
20.1(1.2)
38.3(5.1)
25.5(1.6)
0.019(0.001)
0.09(0.004)
In contrast, the SAND method we proposed consistently outperforms all other approaches, yielding the lowest MSEs and TVs across all scenarios. Table [refeeesss that SAND effectively reduces MSEs and TVs by an average of around 8% and 13%, respectively, compared to GTT. This is further substantiated by Figure which visually demonstrates the smoothness Within the domain of transformer based
SAND-gennraal outputs,
offering empirical support for the validity of Theorem I models, our focus not solely on reducing MSE but also on improving the smoothness of imputations. For this end, we recommend examining the SEs of TVs among different transformer-based models. Upon reviewing TV columns Table [ becomes evident that SAND significantly outperforms all other transformer imputation methods. This highlights the effectiveness of SAND in enhancing the smoothness of imputations. Furthermore, SAND achieves smaller TVs compared to PACE and FACE, suggesting its capacity to capture underlying processes while maintaining low prediction errors.
5.2 Application to real datasets
UK household electricity dataset. We consider a dataset that records every half-hour the energy consumption of 5,494 randomly selected households in London. This results in a total of 96 observations from 1 1/13 (Wed) to 11/14 (Thurs) of 201 See (20) for details of the dataset. Our objective is to recover the underlying process of households' electricity usage where the underlying process is the Gaussian 6-hour moving average of the complete data. To align with our simulation designs, we randomly select the data into n, = 30, n, = to 12, or n, 3, 4, 5 and we sample the time points in two ways independently and dependently. Tables ||andS1]] report that SAND consistently produces the best performance across all settings. When the observed data contain errors, SAND excels in generating smooth trajectories, as illustrated in Figure S2 They also reveal substantial reduction in TVs when comparing SAND to GT in all cases. This shows the effectiveness of SAND in enhancing the smoothness of imputations.
F Framingham heart study. We apply SAND and the baseline methods to the longitudinal data from the Framingham Heart Study [5]. We focus on n 890 subjects who were live at age 70. Our goal is to reconstruct BMI trajectories from ages 40 to 60 based on their irregular and sparse measurements within this range (n: = 3 to 11). Here the MSE (marked by asterisks in Table 2 are measured based on the observed data. Figure SS shows the imputations of two testing data from SAND and its main competitors (PACE, FACE, CNP and GT) and Table 2 highlights the substantial advantage of employing SAND, which reduces MSE by at least 75% for all competitors except for GT. Here GT has lower MSE possibly due to overfitting as reflected in its TV and Figure S3 Regarding TV, which represents the total variation of the imputation in this analysis, PACE, FACE and CNP have smaller TVs than GT and SAND. Figure S3 indicates that they might undersmooth the trajectories. Considering both MSE and TV, SAND emerges as the most robust model across various scenarios.
5.3 Advantage of SAND: Enhancing Downstream Prediction Tasks
This section demonstrates better imputations lead to better downstream predictions in two scenarios: (1) predicting the average energy consumption on November 15, 2013 based on the imputed trajecto- ries of energy consumption of the previous two days, (2) forecasting the average BMI between age 61
During this period, the number of households that had complete observations was maximized.

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and 65 in the Framingham Heart Study using the imputed trajectory of the same subject between age 40 and 60. To predict those outcomes, we leverage AdaFNN (2]], machine learning model capable of learning optimal basis representations between the imputations and the targets. Table 3 shows the result of SAND along with the three top competitors in imputation tasks, PACE, FACE and GT. It's evident that SAND consistently achieves the lowest MSE across all prediction tasks and the principle that better imputations yield more accurate downstream predictions. These findings position SAND as promising tool for enhancing downstream prediction tasks across diverse domains.
Table 3: MSE(SE) on downstream tasks. Bold font marks the smallest MSE across methods.
UK electricity
Framingham study
T1 =
74 = to
n4 = 3, 4,5
ny = to
PACE
5.77(0.9)
11.8(2.2)
15.5(2.9)
2.23(0.7)
FACE
7.23(1.4)
10.8(.1))
13.7(2.))
2.10(0.6)
GT 6.22(1.1)
8.01(1.3)
12.0(1.9)
2.31(0.6)
SAND
5.19(0.6)
7.39(1.1)
12.0(2.0)
1.76(0.5)
6
Discussion
This paper introduces novel SAND layer in transformer for imputing sparse and noisy functional data. SAND, when paired with vanilla transformer, constitutes end-to-end learning mechanism that involves two main operations: Diff 3 and Intg (4). Initially. Diff takes the noisy imputed function from vanilla transformer and concentrates the sub-derivative of the imputation. Following this, Intg constructs differentiable imputation, utilizing the fundamental theorem of calculus. SAND minimizes the distance between the imputed and the original transformer-estimated functions, eliminating the need to estimate the derivative of a noisy function.
From theoretical perspective, we prove the smoothness of the imputation from SAND and explicitly determine the necessary number of hidden nodes for SAND to achieve an prediction error boundary on training data (up constant). Interestingly, the error boundary serves the role of the fraction of variance unexplained in FPCA literature. Traditional imputation methods, such as FPCA, em- ploy dimension-redueee reduction strategy learn trajectories via miite-dimensional vector. Empirical benchmarks reveal that SAND surpasses FPCA when the curves aren't 't a Gaussian process with low number of basis function. SAND effectively smooths noisy imputations from a vanilla transformer, and yields more accurate imputations than other techniques to encourage smoothness.
7 Acknowledgments
We would like to thank the reviewers for their constructive feedback. The research of Jane-Ling Wang and Ju-Sheng Hong was supported by NSF DMS 22- 22-10891 and NSF 24-13924
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Reviews:
[{"review_id": "gWW96FYNJi", "reviewer": "Reviewer_92dE", "summary": "This paper proposes a new class of transformers for sparse and noisy functional data. In particular, a new module, namely self-attention on derivatives (SAND), is incorporated vanilla transformers to model the sub-derivative of the imputed curve, thereby promoting smoothness. The authors also theoretically prove the number of hidden nodes needed by the SAND transformer to achieve a certain prediction error bound for functional imputation. Empirical results are provided to justify the advantages of the proposed model.", "questions": "1. Can you please elaborate more on the use of attention in SAND? Why do we need attention there?\n\nAfter Rebuttal\n\nI will increase my score from 4 to 5.", "limitations": "The authors have adequately addressed the limitations.", "rating": 5, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. The proposed method, i.e., SAND, is interesting.\n\n2. Theoretical properties of SAND is well-studied.  \n\n3. The paper is well-written with illustrative figures.", "weaknesses": "1. Experiments of larger-scale benchmarks are needed to justify the advantages of SAND.\n\n2. Efficiency analysis of SAND is missing.\n\n3. The advantages of using attention in SAND is not clearly discussed in the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the valuable and positive feedback, which we have used to strengthen our paper.  Please see our responses to the weakness below. Our responses to the questions are listed in the global author response.\n\n### [Weakness 1]\n**Answer**: \nThank you for highlighting this aspect. We have not scaled up the simulation for three main reasons:\n- **Simulation Constraints**: As detailed in line 267, each simulated case requires one day for processing on an Nvidia GeForce GTX 1080 Ti. With a total of 88 cases outlined across Tables S2-S10 in the supplementary material, completing all simulations under these conditions requires nearly three months. Even with early-stopping techniques, it still takes up to two months to complete the simulation. These time constraints make larger simulations impractical under our compute budget.\n- **Relevance to Real-World Data**: The sample size of 10,000 in our simulations is substantial, particularly when compared to real datasets we have used, such as the UK electricity dataset with 5,600 samples and the Framingham Heart Study with 870 samples. Simulating data at a scale comparable to real-world datasets ensures that our findings are both realistic and applicable.\n- **Effectiveness of SAND at current scale**: Despite perceptions that 10,000 samples may not seem extensive, our simulations demonstrate a significant advantage of SAND over nine competitors. This indicates robust performance even at the current scale, supporting the effectiveness of SAND on datasets of similar size to popular real-world functional imputation tasks.\n\n### [Weakness 2]\n**Answer**: Thank you for highlighting the importance of providing a detailed efficiency analysis. SAND modifies the conventional attention module by omitting the softmax operation, thereby aligning its computational complexity with that of the original attention mechanism, as detailed in Table 1 of Vaswani et al. (2017).  Let $m$ denote the number of observations in a subject, $h_d$ be the representation dimension, and $k$ be the kernel size of convolutions. We summarize the computational efficiency of SAND as follows:\n- **Compared to Recurrent Modules**: SAND requires $O(1)$ sequential operations, regardless of sequence length, facilitating faster computation and better suitability for parallel processing. In contrast, recurrent layers inherently require $O(m)$ sequential operations due to their dependency on previous outputs for current computations, which significantly slows down processing and limits scalability.\n- **Compared to Convolutional Modules**: The computational complexity of SAND is O(m^2\\cdot h_d), whereas for convolutional modules, it’s $O(k\\cdot m\\cdot h_d^2)$. In our simulation and data applications, the number of observations per subject $m$ is at most 30 while the representation dimension is 128. Given these parameters, convolutional operations become computationally intensive, leading to slower performance compared to SAND.\nSAND, analogous to a single attention module, introduces minimal additional computational overhead compared to standard transformers, which typically comprise hundreds of alternating layers of attention and feed-forward modules. We acknowledge the significance of this aspect and our revised paper clearly articulates the computational complexity of SAND as well as its computational advantages.\n\nRegarding SAND’s prediction efficiency (in the sense of statistical estimation): Our evaluation of SAND’s prediction efficiency, measured through relative MSE, is detailed in Table 1. The results indicate that SAND is approximately 9% more efficient than a standard transformer and 11% more efficient than the best non-transformer benchmark, which is PACE. This demonstrates not only SAND’s computational advantages but also its superior accuracy in predictive tasks.\n\nThis multi-dimensional approach to assessing efficiency—encompassing computational, estimation, and prediction aspects—provides a robust evaluation of SAND’s performance across various metrics.\n\n### [Weakness 3]\nThank you for pointing out the need for a clearer discussion on the benefits of incorporating the attention mechanism in SAND. Here are the key advantages:\n- Performance Enhancement: In section 5.1, our extensive experiments demonstrate that SAND significantly improves imputation accuracy, as evidenced by reductions in mean square error and total variation compared to standard methods. This performance boost underscores the effectiveness of the attention-based approach in handling functional data.\n- Addressing Non-linearity: Traditional post-processing techniques (see Table 1 in the manuscript) in functional data analysis, such as PACE, are typically linear and may fail to capture complex, non-linear features in the data (lines 101 to 102 in the manuscript). SAND effectively bridges this gap by utilizing the non-linear modeling capabilities of the attention mechanism, allowing for a more nuanced and powerful analysis of functional data.\n- Computational Efficiency: Leveraging the attention module within SAND capitalizes on the computational efficiencies discussed in response to [Weakness 2] and as detailed by Vaswani et al. (2017).\nOur revised paper more clearly lists these advantages of utilizing the attention module in SAND.\n\n### [Question 1]\n**Answer**: Please see our response to the integrated question in the global author rebuttal for this question.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "cBQwnMENZA", "reviewer": "Reviewer_pcLe", "summary": "This paper studies the problem of how to perform imputation of the underlying function from noisy or sparse observations with functional data. In particular, the authors present \"SAND\" (Self-Attention on Derivatives), a variant of the transformer architecture, by introducing  $\\mathrm{diff}(\\cdot)$ (derivative) and $\\mathrm{Intg}(\\cdot)$ (integral) operators to the standard transformer, to address the imputation of sparse and noisy functional data. The authors provide theoretical guarantees for the proposed SAND method as well as empirical evaluations across various datasets demonstrate that SAND outperforms existing methods.", "questions": "- The notation in Section 3.1 and Section 3.2 is a bit unclear to me. For example, the bolded tilde $T$ and $T_c$ are $(1+p)\\times M$ matrices, I guess there are positional embeddings inside bolded tilde $T$ and $T_c$. However, these are not formally defined. I would suggest add a notation paragraph to clarify the notation somewhere (could be in appendix in the main body has limited space).\n- In Line 184, why subtracting the first element (or why constructing the tilde $T_c$)? I guess it is because the $\\mathrm{diff}(\\cdot)$ and $\\mathrm{Intg}(\\cdot)$ operators. Would removing this shift term affect the performance of SAND?", "limitations": "NA", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The proposed method with the two ingredients, $\\mathrm{diff}(\\cdot)$ and $\\mathrm{Intg}(\\cdot)$, in the modeling part are well-motivated for tackling the smoothness issue.\n\n- Empirically the paper demonstrates that the proposed SAND transformer can achieve strong performance on both synthetic and real-world datasets.", "weaknesses": "- The theoretical results seem only justify the proposed architecture can approximate the FPCA, which do not provide insights on why this new transformer variant is better than the standard transformer for solving the imputation problem. \n- The design of the $\\mathrm{diff}(\\cdot)$ operator is not very intuitive, since the operator itself does not align well with the derivative operator.\n- I would suggest add a simple baseline (standard transformer with some smoothing post-processing) for comparison. For example, applying certain smoothing technical on top of the output of standard transformers, e.g., locally adaptive regression splines (LARS, Mammen & van de Geer, 1997), trend filtering (Tibshirani, 2014).\n\n[Mammen & van de Geer, 1997] Enno Mammen and Sara van de Geer. Locally adaptive regression splines. The Annals of Statistics, 25(1):387–413, 1997.\n[Tibshirani, 2014] Ryan J Tibshirani. Adaptive piecewise polynomial estimation via trend filtering. The Annals of Statistics, 42(1):285–323, 2014.\n\nI am happy to raise my score if some of the questions/weaknesses are addressed during rebuttal and discussion.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the valuable and positive feedback, which we have used to strengthen our paper.  Please see our responses below.\n\n### [Weakness 1]\n**Answer**: Thank you for the comment. We acknowledge that most of our theorems focus on SAND. However, our theorem does reveal one reason that SAND is better than the standard Transformer:  See Corollary 1 and lines 223 to 226 of the manuscript, which explain how SAND effectively separates the smooth underlying process from random noise by limiting hidden nodes in the network. The standard Transformer is incapable of doing so due to its universal approximation property and the existence of noise in the training data (lines 149 to 154 in the manuscript). When the standard Transformer encounters noisy testing data, they often continue producing similar zigzag patterns, as they have learned to replicate the noise from the training phase.\n \n### [Weakness 2]\n**Answer**: The traditional numerical derivative operator takes the form $f^\\prime(a) \\approx \\frac{1}{b - a}[f(b) - f(a)]$, a linear combination of the inputs $f(a)$ and $f(b)$ with fixed weights (line 195). Our $\\rm{diff}(\\cdot)$ operator similarly calculates a weighted sum of these inputs, with weights learned by the network.\nOur decision to design the $\\rm{diff}(\\cdot)$ operator to resemble the attention module was driven by two main considerations:\n- **Cohesion with the Transformer Architecture**: Our SAND module is a simple  augmentation of the standard Transformer. SAND modifies the conventional attention module by omitting the softmax operation, thereby aligning its computational complexity with that of the original attention mechanism, as detailed in Table 1 of Vaswani et al. (2017). Compared to recurrent and convolutional networks (other architectural techniques for achieving smoothing), SAND enjoys major computational advantages (Vaswani et al., 2017).\n- **Leveraging Proven Mechanisms:** The attention mechanism has demonstrated effectiveness across numerous tasks. By minimally adapting this component, we aim to preserve this versatility in our operator. The $\\rm{diff}(\\cdot)$ operator retains the core benefits of attention while extending its application to approximate derivatives and enforce smoothness.\n\n### [Weakness 3]\n**Answer**: We have indeed considered the approach of post-processing imputations from a standard Transformer. As discussed in lines 164 to 168 and in Table 1 under the rows labeled “GT1P”, “GT1S”, “GT2P”, and “GT2S”, we applied both PACE and kernel smoothing as post-processing methods to imputations from a standard Transformer model.\nThe effectiveness of these methods is discussed in lines 302 and 307, where we note that while the total variation of the estimated function generally decreases, the improvement in MSE is marginal, and in some cases, the MSE actually increases after post-processing. These findings underscore the limitations of simply adding some smoothing post-processing to standard Transformers, regardless of which smoothing technique is used.\nWe chose PACE as a popular imputation method for noisy functional data, and we chose kernel smoothing due to its simplicity and historical prominence as the canonical  smoothing method. Following your suggestion, we have also applied trend filtering to the output of standard Transformer, by using  `trendfilter` from the package `genlasso` in `R`. The results are in the PDF file attached to our global rebuttal. Although the improvement is greater than using PACE or kernel smoothing as post-processing, SAND remains the best overall method. Our revised paper incorporates and discusses these results, as well as accounting for the other valuable feedback.\n \n### [Question 1] \n**Answer**: Yes, there are positional embeddings inside bolded $\\tilde{T}$ and $\\tilde{T}_c$. The notation is defined in lines 185 and 186: bolded $\\tilde{T}$ (or $\\tilde{T}_c$) are matrices with the first row being $T$ (or $T_c$), and the rest of $p$ rows are the positional encoding of the output grid. Thank you for the suggestion. Due to the limited space in the main text, we will add a notation paragraph to in the appendix.\n \n### [Question 2]\n**Answer**: The subtraction of the first element serves multiple purposes:\n- **Geometrical Justification**: This adjustment ensures that two curves, which may have identical shapes but different intercepts (the first element in $T$), are treated equivalently by the $\\rm{diff}(\\cdot)$ operator. By subtracting the first element, we effectively remove the absolute positioning, focusing solely on the shape of the curve.\n- **Mathematical Rationale**: Our approach is inspired by the first fundamental theorem of calculus, which expresses a function as $f(b) = f(a) + \\int_a^b f’(x),dx$ (refer to line 201 in the main text). In the context of SAND, $f(a)$ represents the first element of $T$, serving as the initial value. The $\\rm{Intg}$ function acts as \"$\\int$\", and $\\rm{diff}$ approximates the derivative. Subtracting the first element aligns with this conceptual framework, where $f(a)$ is reintegrated post-differentiation to reconstruct the curve accurately (see Equation 5 between lines 189 and 190).\n- **Algorithmic Efficiency**: Subtraction normalizes the input data, enhancing the uniformity of the inputs. This normalization can lead to faster convergence of the $\\rm{diff}(\\cdot)$ operator’s parameters during training.\nOmitting the shift could potentially require more training epochs for SAND to achieve similar convergence, due to the variability in initial values across data samples.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "a9GRgEqS2j", "reviewer": "Reviewer_zTxC", "summary": "The paper addresses the limitations of ordinary transformers for doing imputation of functional data over irregularly longitudinal data. The authors propose a novel new variant of transformer that takes derivatives into account, called SAND. Theoretically, it's shown that SAND with a certain number of hidden neurons can do functional imputation well. Then empirically, extensive experiments show that SAND perform much better than other methods.", "questions": "No.", "limitations": "Yes, limitations are addressed adequately and no potential negative societal impact.", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "1. Clear writing. Everything is explained quite well, and the flow of logic is smooth. It's immediately clear to me that the authors study an important problem and do well.\n2. A complete story with theories and empirical results. The paper uses a lot of math, but it's used quite properly, because the mathematical comes naturally out of the structures of the problem itself, rather than some artificial assumptions. The theoretical analysis properly justifies the design the new transformer architecture and is very neat. Then there are extensive empirical studies that prove the usefulness of the new architecture.\n3. The authors study a specific class of problems with great clarity. I'm so tired of papers claiming to have one method that improves things in great generality, which usually are not evaluated completely. It's good to see that we have steady progress in developing solid tools for specific problems of certain structures.", "weaknesses": "I'm not seeing effective weaknesses.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the valuable and positive feedback.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We are grateful for the detailed feedback from all reviewers, which has significantly contributed to refining our manuscript. We would like to address a specific concern raised by two reviewers (**92dE** and **pcLe**) regarding the use of attention in SAND's $\rm{diff}(\cdot)$ operator and the design of it. These concerns relate to fundamental aspects of our methodology and its innovative application in functional data imputation.

### [Integrated Question] Can you elaborate on the conceptual foundations and decision-making process behind the integration of the attention mechanism and the design of the $\rm{diff}(\cdot)$ operator in SAND?
**Answer**: We arrived at our decision to use the attention module in SAND as follows:  We initially started with a coarse imputation from a standard Transformer and aimed to refine its poor performance. One intuitive method could involve adding penalties to the standard transformer to constrain its outputs or applying posterior smoothing techniques to enhance the coarse imputation. We discuss the limitations of these approaches in lines 156 to 169 and benchmark their empirical performance in section 5.1.
An alternative approach is to apply a “patch” to the standard transformer such that the resulting architecture outputs a smooth imputation, a process depicted in Figure 2. Conceptually, we could utilize any machine learning model that handles vector inputs and outputs to achieve this, as there are numerous viable options.
We adapt the attention module as our patch in SAND for several reasons:
- Computational Efficiency: SAND is computationally efficient compared to other well-established vector-to-vector models like recurrent networks or convolutional networks. This efficiency is crucial in our choice, as highlighted by Vaswani et al. (2017). The attention module itself has demonstrated effectiveness across diverse applications, thanks in great part to its scalability.
- Achieving Smooth Outputs: Our objective is to produce a smooth curve, which requires that its first derivative be continuous. Since most neural networks inherently model continuous functions, they are suited to this task. The first derivative $f^\prime(a) \approx \frac{1}{b - a}[f(b) – f(a)]$ involves a linear combination of function values. In constructing SAND, we chose the attention module because it inherently performs a weighted summation of its inputs, aligning well with our needs for modeling derivatives (as detailed in Equation 1 and lines 130 to 134 in the manuscript).

These considerations guided our decision to employ the attention mechanism in SAND, which enjoys favorable theoretical properties and empirical performance.

We provide individual responses below to address the remaining concerns from each reviewer to improve clarity of missing details and to provide additional discussion that strengthen our paper. We thank all reviewers’ for their time and efforts! We hope our responses have persuasively addressed all remaining concerns. Please don’t hesitate to let us know of any additional comments or feedback on improvement.

Note that we include all additional experimental results in the one-page pdf submitted along with this global rebuttal response.

Meta-review: All reviewers were in agreement that this paper should be accepted, as it develops a well motivated method and especially due to the strong experimental results. However I strongly urge the authors to address the concerns that came up during the process. While I cannot summarize everything that came up, here is one issue that the authors should address: Larger scale experiments should be done (the Nvidia GeForce GTX 1080 Ti that the authors refer to is by now vastly outdated--- it was released in 2017, before the Attention mechanism!).
Decision: Accept (poster)