Learning Causal Alignment for Reliable Disease Diagnosis
Authors: Mingzhou Liu
Ching-Wen Lee
Xinwei Sun
Xueqing Yu
Yu QIAO
Yizhou Wang
Venue: ICLR 2025 Poster (2025)
Keywords: alignment
causal learning
counterfactual
disease diagnosis
Primary area: alignment, fairness, safety, privacy, and societal considerations
TL;DR: We propose a casual alignment loss for aligning the decision-making process with physicians

Abstract: Aligning the decision-making process of machine learning algorithms with that of experienced radiologists is crucial for reliable diagnosis. While existing methods have attempted to align their prediction behaviors to those of radiologists reflected in the training data, this alignment is primarily associational rather than causal, resulting in pseudo-correlations that may not transfer well. In this paper, we propose a causality-based alignment framework towards aligning the model's decision process with that of experts. Specifically, we first employ counterfactual generation to identify the causal chain of model decisions. To align this causal chain with that of experts, we propose a causal alignment loss that enforces the model to focus on causal factors underlying each decision step in the whole causal chain. To optimize this loss that involves the counterfactual generator as an implicit function of the model's parameters, we employ the implicit function theorem equipped with the conjugate gradient method for efficient estimation. We demonstrate the effectiveness of our method on two medical diagnosis applications, showcasing faithful alignment to radiologists.

Published as a conference paper at ICLR 2025
LEARNING CAUSAL ALIGNMENT FOR RELIABLE DISEASE DIAGNOSIS
Mingzhou Liu' Ching-Wen Lee' Xinwei Sun Xueqing Yu' Yu Qiao Yizhou Wang41.867
1 School of Computer Science, Peking University
2 School of Data Science, Fudan University
3 School of Automation and Intelligent Sensing. Shanghai Jiao Tong University
4 Center on Frontiers of Computing Studies, Peking University
5 Institute for Artificial Intelligence, Peking University
6 Nat'l Eng. Research Center of Visual Technology, Peking University
7 State Key Lab. of General Artificial Intelligence, Peking University
* Corresponding authors
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ABSTRACT
Aligning the rrcision-making process of machine learning algorithms with that of experienced radiologists is crucial for reliable diagnosis. While existing meth- ods have attempted to align their diagnosis behaviors to those of radiologists re- flected in the training data, this alignment is primarily associational rather than causal, resulting in pseudo-correlations that may not transfer well. In this paper, we propose causality- based alignment framework towards aligning the model's s decision process with that of experts. Specifically, we first employ counterfactual generation to identify the causal chain of model decisions. To align this causal chain with that of experts, we propose a causal alignment loss that enforces the model to focus on causal factors underlying each decision step the whole causal chain. To optimize this loss that involves the counterfactual generator as an im- plicit function of the model's parameters, we employ the implicit function the- orem equipped with the conjugate gradient method for efficient estimation. We demonstrate the effectiveness of our method on two medical diagnosis applica- tions, showcasing faithful alignment to radiologists. Code is publicly available at https::/gi com mz1 23321 / aliggment
INTRODUCTION
Alignment is essential for developing reliable medical diagnosis Systems/huang & Hadfield-Menell 22020). For instance, in lung cancer diagnosis, using models that misaligned with clinical pro- tocols can result in reliance contextual features or instrument markers (Fig. (c) for diagnosis, leading to misdiagnosis and loss of timely treatment.
Despite the importance, alignment in medical imaging systems is largely understudied. Existing studies that are mostly related to primarily focused visual alignment, including Zhang et al. (2018;; Chen et al. (2019); Brady et al. (2023) that proposed to learn object-centric representations, Hind et al. (2019) 9); Rieger al. (2020) that adopted multi-task learning schemes to predict labels and expert decision bases simultaneously. Particularly, recent works Ross et al. (2017); Gao et al. within expert-annotated areas. However, their alignment with expert behaviors is only associational, rather than causal, making their models still biased towards spurious correlated features. This limitation is further explained in Fig. (a), where two decision chains with different causal structures exhibit similar correlation patterns.
2022): ; Zhang et al. (2023) have proposed to regularize the model's input gradient
In this paper, we propose causal alignment approach that focuses the alignment in the underly- ing causal mechanism of the decision- maaiiinnmmminggggg process. Specifically, we first identify causal factors behind each decision step using counterfactual generation. We then propose causal alignment loss

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Aligned (X' ~ A - F)
Misaligned (shortcut-+ V)
F
Caleification Subelity
Benign X
Margin v
Maligmantt V
Spiculation
Causal Alignment
shortcut - +
Caleiftration
fo
Subilety Margin V
Benign Malignant
(a)
Similar Correlation Patterns
Spiculation
Input
Mixaligned
Aligned [Duns]
Pestidon CAM Doteart)
(c)
Figure 1: (a) Two decision chains with different causal structures but present similar correlation patterns. The left chain "mass (X) attributes (A) - label (Y"" aligns with radiologists, while the right chain "shortcut + label (Y)" is misaligned. However, both (X, Y) and (shortcut, Y) are correlated, due to the confounding bias between the shorteut and Y. (b) Our approach for learning causally aligned models. We first identify features and attributes that causally influence the model's decision, then align them to that of radiologists in a hierarchical fashion. (c) Class Activation Map- ping (CAM) visualization and comparison of CAM precision on lung cancer diagnosis.
to enforce these identified causal factors to be aligned within those annotated by the radiologists. To optimize this loss that involves the counterfactual generator an implicit function the model parameters, employ the implicit function theorem equipped with the conjugate gradient algo- rithm for efficient estimation. To illustrate, we consider the lung cancer diagnosis as shown in Fig, [ (b). Guided by the alignment loss, our model can mimic the clinical decision pipeline, which first identifies the imaging area that describes attributes of the lesion, and then diagnoses based on these attributes Xie et al. 22020). Such training facilitated by employing causal attribution Zhao et al. (2023) for inferring attributes that are causally related to the diagnosis. Returning to the lung cancer diagnosis example, Fig. (c) shows that method can learn causally aligned representations, in contrast to the features adopted by baseline methods, which are challenging to interpret.
Contributions. To summarize, contributions
1. (Causal alignment) We propose novel causal alignment approach to achieve alignment of causal mechanisms underlying the decision process of experienced radiologists.
2. (Optimization) We propose efficient optimization algorithm by employing the implicit function theorem along with the conjugate gradient method.
3. (Experiment) We demonstrate the utility of our approach through significant improve- ments in alignment and diagnosis, lung cancer and breast cancer diagnosis tasks.
RELATED WORKS
Learning visual alignment. Alignment is more broadly studied, e.g., in natural language processing
Ouyang al. (2022) and reinforcement learning Ibarz et (2018). In the realm of visual alignment,
Hind et al. (2019): Rieger et al.) (2020) proposed to align deep learning models with humans by si-
multancously predicting the class label and the decision area. Zhang et al. (2018) 8): Liu et al, Ib);
Müller al. (2023) aligned the decision-making process of neural networks by incorporating
pert knowledge into architecture design. Of particular relevance to our work are Ross et al. (2017);
Gao et al. (2022); Zhang et al. (2023). which suggested constraining the input image gradient to be sigmificant in areas annotated by experts. However, the input gradient can be biased by pseudo-
correlations that exist between expert features and shortcut features Geirhos et al. (2020) leading
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to misaligned model. In contrast, adopt counterfactual generation to identify causal areas that determine the model's prediction. By ensuring these factors are confined to model can be effectively aligned with the expert's 's decision process.
areas,
Explaining medical AL. Explainability is essential for physicians to trust and utilize medical diag- nosis models Lipton 2017). To achieve this, aftrübution-busee methods explained model predictions by assessing the importance of different features Suryani et al. (2022); Yuen (2024). Example-bate based methods utilized similar images Barnett et al. (2021) or prototypes Gallée al. (2024) to interpret the underlying decision rules. However, these approaches focused on interpreting models that have been trained. If misalignment occurs during the training process, their utility could be limited. In contrast, we propose an alignment loss to learn an intrinsically explainable model.
3 PROBLEM SETUP & BACKGROUND
In this section, we formulate our problem and introduce the background knowledge.
Problem setup. We consider the classification scenario, where the system contains an image x X and a label E y/ from an expert annotator. In addition to y. we assume the expert also provides an explanation e to explain his decision of labeling as y. Commonly, the explanation could refer to region of interest annotations or attribute descriptions. For example, radiologists often write an annotation section and an observation section, which respectively describe which body part is abnormal and what phenomena are observed, in their reports to explain their diagnosis Xie et al. (020). Motivated by this, we assume for each sample, the explanation can be formulated as a binary mask m. indicating the abnormal area, along with a binary attribute description a = [01, ..., Up) E A of the abnormality. In this regard, our data can be denoted as DD {( Dis C4 = (m, 0, 1=-1- With this data, our goal is then to learn classifier fa X' + } that i) predicts y accurately ii) has a decision mechanism that is aligned with the radiologists.
Structural counterfactuals. To measure the likelihood that one event caused another, Pearl 2009) defines the following counterfactual quantity known as the probability of causation:
P(Y, v Tn, Y = yo):
which reads as "the probability of Y would be y had been x: if we factually observed that X = 2o and Y = you [ Here, Yz denotes the unit-counterfactual |Pearl (2009) or potential outcome. In our scenario, rather than considering the whole image x, we are interested in specific regions within the image that causally determine the model's decision. To identify these regions, we adopt the following counterfactual generation scheme.
Counterfactual (CF) Generation. Given the classifier fe and any sample pair (zo, y0). CF gen- crates the counterfactual image x² with respect to the counterfactual class y² * y via Dhurandhar сt al. (2018;; Verma et al. (2020); Guyomard et al. (2023) Augustin et al. (024);
x²' arg min Lee (f v + ad(x,,
where Lee is the cross-entropy loss for classification, d(:,-) is distance metric that constrains the modification to be sparse, and a is the regularization hyperparameter. In this regard, the modified area supp(x* - wo) is responsible for the classification of xip as yo. in that if we modified xo to x², the model would have made different decision y². Indeed, in Prop. A.5 we can show that xt
maximizes the probability of causation (Y, y² X = xo. Y yo! induced by the classifier so. subject to d(z, z0) da for some dy.
To ensure the realism of the generated image, we can implement (2) using gradient descent in the image's 's latent space. Notably, this approach has proven effective for generating realistic images Goyal et al. 2019); Balasubramanian et al. (2020); Zemni et al. (2023), as also verified by the visualization of generated counterfactual images in Fig.
Indeed, 22) is similar to but different from the optimization in Adversarial Attack (AA) Szegedy et al. (2013) concerning perceptibility Verma et al.) (2020). Although both methods share the same objective framework, CF aims at highlighting significant areas that explain the classifier's 's decision
UUnde the exogeneity and monotonicity conditions for binary X, Y, this quantity is identifiable. "This term identifiable since fo is known (see Prop. A.4 for details).
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Input
Attributes Prediction
Classification
loss Uo0tly)
Caleifioction
fa
barabied taarsform srage grodent
Subtletyy
Benign x
for
foz
Margin
Malignanc v
Inpii Giadler Sihkr
Spiculation v/
[by] P.Ref
Fangen
m)
(a) Hierachical Architecture
(b) Alignment Loss
Figure 2: The schematic overview of our method. (a) We adopt a hierarchical structure that first provides attribute descriptions for the image, and then shows the diagnosis result. (b) Training with the proposed alignment loss. In the forward pass, counterfactual image generated and used to compute the alignment loss Lalign relative to the expert's annotation m. In the backward pass, we use an implicit gradient solver obtain the gradient Volalign and use update the parameter
process, whereas AA favors making small and imperceptible changes alter the prediction outcome Wachter et al. (2017). This often leads to different choices of the distance function d(-; and the hyperparameter Freiesleben (002); Guidotti (2024).
METHODOLOGY
In this section, introduce our framework for medical decision alignment. This section is com- posed of three parts. First, in Sec. [. we introduce a causal alignment loss based on counterfactual generation, to align the model's decision bases with those of experts. Then, in Sec. 4.2 propose to use the implicit function theorem equipped with conjugate gradient estimation to compute the gradient of our loss for optimization. Finally, in Sec. [.33 we enhance method with hierarchical alignment for cases where attribute annotations are available, using hierarchical pipeline based on causal attribution. We summarize framework in Fig.
4.1 CASUAL ALIGNMENT LOSS
In this section, we propose causal alignment loss to align the model with the experts. For illustra- tion, we first consider the case where the attribute annotations are unavailable. The idea of our loss is to penalize the model once its counterfactual image contains modifications beyond radiologist- annotated areas. Specifically, we optimize a loss Lalign of the following form:
n
Latign Σ
(( xz) (1-m))lls
where denotes the element-wise matrix product, x; the counterfactual image x; obtained by 2). and m {0, 1] dim(=x) is the binary mask provided by radiologists. Then, by combining fatign with the cross-entropy loss for classification, we have our overall training objective:
Lee + A.alignn
where À is a tuning hyperparameter. To understand how the objective works towards alignment, note that x maximizes the counterfactual likelihood Pa (Yz = y (ro, Wo), indicating that supp(z" - wr) represents the causal factors that influence the decision of the model fo. Therefore, minimizing the distance between cupp(x* ro) and m encourages the model's causal factors to align more closely to those of the experts.
Our loss enjoys several advantages over alternative methods in visual alignment. Compared to Liu et al. (2021al b); Müller et al. (2023) that incorporated prior knowledge into network architectures, loss is flexible and be easily adapted other scenarios and backbones. In contrast to Ross et al. (2017); Zhang et al. (203) that constrained the input gradient, our approach can effectively avoid pseudo-features, benefiting from the identification of causal factors.
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4.2 OPTIMIZATION
In this section, we introduce the optimization process for the proposed alignment loss. For opti- mization, we need to compute the gradient VaLalign: which involves the Jacobian matrix Vor** The main challenge here is that x" is an implicit function of 0, defined by the argmin operator in 2 which makes it hard compute Vaz" explicitly.
To address this challenge, we resort to the Implicit Function Theorem (IFT), which allows us to compute the gradient an implicit manner. Specifically, note that if x is the minimum point of the function T(x, 0) = (fo(f(()),)) v") + ad(2, zo). it should satisfy that:
O.
According to the law of total derivation, this implies that:
V (V2(V.T) Ver" Vo((V =0.
Therefore, computing V px boils down to the problem of solving the following linear equation:
H
where we denote H == TV) as the Hessian matrix, 2" == Vex" as the Jacobian matrix, and b VV(V,T) as the negative mixed derivative for brevity.
Formally speaking, we have the following theorem:
Theorem 4.1 (Implicit Function Theorem (IFT), Krantz & Parks (2002)). Consider two vectors r. O, and differentiable function T(x, 0). Let x² arg min, T(x, 0). Suppose that: i) the argmin is unique for each 0, and it) the Hessian matrix H invertible. Then x² (0) is a continuous function of O. Further, the Jacobian matrix V exx satisfies the linear equation
Thm. 4.1 suggests that we can compute the Jacobian matrix using Vor" = -H which then gives Lalign with the chain-rule. Nonetheless, for imaging tasks, typically 0 is the parameter of high-dimensional neural networks, making it intractable to compute the Hessian matrix and its inverse. To address this issue, we employ the conjugate gradient algorithm| Vishnoi et al. (2013) to estimate the solution of (4), without explicitly computing storing the Hessian matrix. Notably, the conjugate gradient method has been successfully deployed in Hessian-Fie -rre methods for deep
learning Martens al. (2010) and meta learning Sitzmann al. 2020).
We briefly introduce the idea of conjugate gradient below, with a detailed discussion left to Vishnoi al. (2013) (Chap. 6). To begin with, note that solving (4) is equivalent solving:
= arg min g(2), where g(2) = 2" HH -
in that the minimum point =* satisfies V=g = - 0.
In this regard, we can implement gradient descent to minimize g(-), where the minimum point gives the solution of (4). During the minimization, the direction of the gradient updating set to be conjugate (i.e., orthogonal) to the residual 6 ( where z(1) is the estimate of 2 in the i-th iteration, in order to achieve optimal convergence rate. To achieve this without explicitly forming H, we can leverage the Hessian vector product Song & Vicente (022). Specifically, for that is small perturbation around 2, we have:
+z(2+ ~ Vg(2) + Hez()
then follows that:
H2( Vg(=+ez - Vg(2)
which means we estimate 02 with the finite difference of Vg the right- hand side.
Equipped with Thm. J.meeecallll the conjugate gradient method for estimation, we now summarize the optimization process for loss in Alg.
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Algorithm Causal alignment training
Input: Data D,
Output: Decision model fe.
Hyperparameters: Sparsity regularization a, weight of alignment loss X, learning rate 17.
1: while not converged do
2: ** **Forward pass
3: Compute Leee
4:
Optimize (2) to obtain x" and compute Calign using
5:
Compute L Lox + Xaalign.
6:
**Back propagation
7:
Estimate V Valulg with conjugate gradient.
8:
Update 0: f + 0 VoL. or Adam
9:
end while
4.3 HIERARCHICAL ALIGNMENT
In this section, we extend our method to the scenario where attribute annotations are available. We introduce a hierarchical alignment framework mimic the clinical diagnostic procedure.
Causal diagram and assumptions. We characterize this diagnostic process with the causal graph in Fig. According McNitt- Gray al. (2077; : Lee al. (217), the first step the diagnosis is annotating each mass attribute from the image Xie et al. (2020). Therefore, we assume causal edges from the image to the attributes A. Since these attributes are directly annotated from the image, we assume no additional dependencies among them, implying their conditional independence given X. Building on these attributes, we further assume causal relationship - Y, representing the eecision-making process from the attributes to the final decision label.
Specifically, our classifier fe consists of an fe, : X + A that predicts the attributes from the image and an fez A - y that classifies the label based on the predicted attributes. For counterfactual generation, for predicting by altering the predicted attributes == Sa, (x) the counterfactual ones a". Then, locate image features that account for the modification of la"-a] via another counter- factual optimization over x and obtain the counterfactual image x² For hierarchical alignment, we require both Ia* al and 1x" x|| to be aligned with the expert's annotations of causal attributes and image regions, respectively.
A
first find attributes responsible
A2
Ap
Causal diagram of radi- ologists' decision process. A and denote the expert's annotations of the attributes and the decision label, respectively.
Figure 3:
Causal attribution for annotations. Although the attribute annotations can be available for many cases Armato III et al. (201T): Lee al. (2017). it hard to know which ones of these attributes causally determined the labeling of radiologists for each specific patient. To identify the causal attributes for alignment, we employ causal attri- bution based on counterfactual causal effect Zhao et al. (2023), which extends ① to enable the quantification of the probability of causation for any subsets of attributes while conditioning on the entire attribute vector. Specifically, given evidence of the attributes A and the label Y y, we calculate the Conditional Counterfactual Causal Effect (CCCE) score for each attribute subset C d.....,ii(AA)))
CCCE(S) = ([ Y = a, Y yy)
which is the difference between the conditional expectations of the potential outcomes Y YA==1 and YA given the evidence. Recall that each attribute A; is binary. Then, according to Zhao et al.
(2023 (Thm. 2), CCCE(S) is identifiable and equals
CCCE(S) ( 11 (YY
A = a) @ 1 P(Y P(Y = y A
As = 1 1,
P(Y = y -
where A_ denotes attributes beyond the subset S. Here, "(1)" arises from the exogeneity condition that there is confounding between A and Y (i.e, Ya 1. A), and "(2)" is based the monotonicity

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condition that Y. < Ye' if a Both conditions natural to hold in scenario. Specif- ically, the exogeneity condition holds since the radiologist's 's decision Y is based only on attributes (Fig. For the monotonicity condition, is easy to that each intervention on any attribute from 0 to (e.g., from no speculation to speculation) will raise the probability of malignancy.
After computing the CCCE score for each attribute subset, select the subset S with the highest CCCE as the set of attributes causally related to the label. Accordingly, we set the annotation vector r E {0, 1) Adim(A) such that rg = (1,..., 1...,.) 1)T.
Hierarchical alignment. With such annotations, we introduce our hierarchical alignment process. Specifically, our objective function over (01,02) is:
((0) == (fol (fe, (x)), v) + (fo (x), - X2Catign (02) + a1 Calign (01),
(5)
where A1 > 0, A2 0 are tuning hyperparameters. Here, (So (fo, (z)), v) and Loo (fo, (x), denote the cross-entropy losses for predicting y and a, respectively.
The alignment loss Latign (02) over O2 is defined as:
a
Lalign( (02)
Σ (aa)(0)) â.) @ (1-11))ll (1-rr)le
where à; : fo. (x;) and the counterfactual attributes a" (02) is generated via: a" (02) arg min L (fez (a'), v²) azd(d', a).
(6)
Similarly, the alignment loss Laiign (01) over 01 is defined by 3 where the counterfactual image x= (01) that explains the change of to is generated by:
x' =01)) = arg min (fe (x') (x'),a") + ajd(r';x))
With the objective 5 optimize 0 by applying Alg. to alignment terms. After the optimization, our decision process x - fe, (fo, (z)) aligns well with that of the experts, with fo, employing causal imaging factors predict attributes, and foz using the causal attributes predict y.
EXPERIMENT
In this section, we evaluate our method two medical diagnosis tasks: the benign/mallgnant clas- sification lung nodules and breast masses.
5.1 EXPERIMENTAL SETUP
Datasets & Preprocessing. We consider the LIDC-IDRI dataset Armato III al. (2011) for lung nodule classification and the CBIS- DDDM dataset Lee et al. (2017) for breast mass classification.
The LIDC-IDRI dataset contains thoracic CT images, each associated with bounding boxes indicat- ing the nodule areas, six radiologist--annotteed attributes (subtlety, calcification, margin, speculation, lobulation, and texture) and malignancy score ranging from to Before analysis, we preprocess the images by resampling the pixel space and normalizing the intensity. We label those images with malignancy scores of 1-3 as benign (y = and those with scores of 4-5 malignant (y 1). We split the dataset into training (n 731), validation (n 238), and test (n = 244) sets. The CBIS- DDSM dataset contains breast mammography images with fine-grained annotations (mass bounding boxes, attributes, and malignancy). We preprocess the images by removing the background and nor- malizing the intensity. We use the provided binary malignancy label and six annotated attributes (subtlety, shape, cireumseription, obscuration, ill-definiteness, and spiculation). We follow the offi- cial dataset split, with 691 masses in the training set and 200 masses in the test set.
To test the ability of our method to learn expert-aligned features, we add "4"/'-" symbol on the top-left corner of each image as a spuriously correlated feature. This symbol coincides with the
Zhho al. (002)) also assumed exogeneity condition and the monotonicity conditions among A, if there exist causal relations among A. Since there are no causal relations among A, we do not need these conditions. 'Here, and a' are both vectors. a < a' denotes as a for each k.
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malignancy label in the training set, where images with y = 1 are labeled with "+" and those with y = 0 are labeled with "-"; ; but are assigned randomly in the validation and test sets. A well-aligned antfologis-amnnttteed model should focus on the radiologist annotated areas rather than the symbol.
Evaluation
metries. To assess the alignment of our model relative to radiologists, we compute the Class Activation Mapping (CAM) Selvaraau al. (0177) and report its precision relative to the annotated areas, i.e.,
Area of (CAM n Anno), We also report the overall classification accuracy. Area CAM
Implementation details. We use the Adam optimizer and set the learning rate as 0.001. We pa- rameterize the attributes prediction network fo, with a seven-layer Convolutional Neural Network (CNN), and train it for 100 epochs with a batch size of 128 for each iteration. For the classification network fog. we parameterize it with two-layer Multi-Layer Perceptron (MLP), and train it for 30 epochs with a batch size of 128. Please refer to Appx. B for details of the network architectures. For the hyperparameters ai in 7 and 022 in (6). we them a 0.01, 02 - 0.0005 for LIDC-IDRI and 01 0.07, 02 0.0005 for FBIS-DDSM, respectively. For both datasets, we set A = X in (5). For causal attribution, we calculate the CCCE scores of subsets containing no more than three attributes and select the subset with the highest score. We adopt the TorchOpt Ren et al. (2022) package to implement the conjugate gradient estimator. We repeat 3 different seeds to remove the effect of randomness.
5.2 COMPARISON WITH BASELINES
Compared baselines. We compare our method with the following baselines: i) Ross et al. (0117) that achieved interpretability by penalizing the input gradient to be small in object-irrelevan areas; ii) ICNN ZZangg et al. (2018) that modified traditional CNN with an interpretable convolution layer to enforce object-centered representations; iii) BagNet Brendel & Bethge (2019) that approximated models; iv) Rieger et al. (2020) that required the model to produce a classification as well as an explanation (i.e., multi-tasks learning): v) "Chang et al. (2021) that augmented the dataset with various factual and counterfactual images to alleviate the problem of learning spurious features; and vi) the Oracle classifier in which we manually restrict the input features to areas annotated by radiologists.
CNN with white-box bag-of-feature
Table 1: Comparison with baseline methods on LIDC-IDRI and CBIS-DDSM datasets. The result of our method is boldfaced and the best result among baseline methods is underlined. For the Oracle classifier, the input features are manually restricted to the areas annotated by radiologists.
Precision of CAM
Classification accuracy
Methodology
LIDC
DDSM
LIDC
DDSM
Ross et al.
2017
0.034 (0.06)
0.084 (0.11)
0.656 (0.00)
0.559 (0.05)
Zhang a.
2018
0,068 (0.11)
0.110 (0.13)
0.381 (0.03)
0.581 (0.00)
Brendel & Bethge
2019
0.048 (0.04)
0.090 (0.04)
0.358 (0.00)
0.592 (0.00)
Rieger et al.
2020
0.041 (0.05)
0.232 (0.17)
0.343 (0.00)
0.586 (0.0T)
Chang ct al.
2021
0.074 (0.03)
0.119 (0.07)
0.503 (0.08)
0.496 (0.08)
Oracle classiier
1.000 (0.00)
1.000 (0.00)
0.789 (0.00)
0.726 (0.01)
Ours
0.751 (0.03)
0.805 (0.06)
0.722 (0.00)
0.656 (0.00)
Due to the capability of capturing causal features, our method also significantly surpasses baseline models in terms of classification accuracy. This is due to the fact that, unlike the "4"/""" symbol that demonstrates only spurious correlation to the label, features within the annotated areas have a causal relationship with the label, and therefore are transferable to test data.
Additionally, it is worth noting from Tab. I that even the oracle classifier only reaches a classifi- cation accuracy of 72% 79%, which seems to contradict some previous results Wu et al. (2018)
Wang et al. 2022): ; Liu et al. (003)) that claimed an accuracy of more than 99% in lung nodule classification and 90% in breast mass classification. To comprehend, this discrepancy is primarily due to the exclusion of challenging samples (those with a malignancy score of 3) in Wu et al. (2018)
and the usage of eustom training/test sets split in Wang et al. (002)); : Liu et al. (2023).
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5.3 ABLATION STUDY
In this section, we perform ablation study the causal alignment loss (Sec. 4.1 chical aligument process (Sec 4.3 The results are shown Tab.
and the hierar-
Table Ablation study on LIDC-IDR and CBIS-DDSH datasets.
Precision of CAM
Classification accuracy
Calign
Hierarchical align
LIDC
DDSM
LIDC
DDSM
0.057 (0.07)
0.143 (0.20)
0.535 (0.08)
0.592 (0.00)
0.587 (0.08)
0.621 (0.03)
0.701 (0.02)
0.633 (0.03)
0.751 (0.03)
0.805 (0.06)
0.722 (0.00)
0.656 (0.00)
As we can see, both the alignment loss and the hierarchical procedure significantly improve the performance. In detail, the alignment loss accounts for a substantial portion of the improvement, yielding 50% increase in CAM precision and a 15% boost in classification accuracy. Additionally, the hierarchical training strategy contributes an extra 20% to aligmment precision and 2% increase classification performance. These results demonstrate the effectiveness of our alignment loss in learning features that coincide with radiologist assessments, as well the significance of the hierarchical training strategy in mimicking the clinical diagnosis process.
5.4 VISUALIZATION
To further verify whether our method can learn radiologist-aligned features, visualize the Class Activation Mapping (CAM) and show the results in Fig.
Input image
BagNets
Rleger
Chaing
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Figure 4: CAM visualization. Each row denotes different cases. The first column is the input images, where nodules and masses are marked by red bounding boxes. The second to seventh columns are CAMs of compared baselines and our method, respectively. See Appx. C.4 for more results.
As shown, the activation our method concentrates on the nodule/mass areas, especially on the margins of the nodules/mass, which is key feature for radiologists to evaluate the malignancy Sandler et al. (003)) Liu et al. (2024). contrast, the activation of baseline methods focuses on fesion-irrelvaat areas, such as the shortcut symbol "4"/"=" region in the top-left corner for Ross et al. (2017) and Brendel & Bethge (2019), or the background areas for Zhang et al. (2018). Rieger al. 2200)) and Chang al. (2021). This visual analysis corroborates the quantitative results, demonstrating our method's ' ability to learn features that are well well-aligned with the radiologist's 's agnostic process.

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6 CONCLUSION AND DISCUSSION
In this paper, we present a causal alignment framework to bridge the gap between the decision- making process of machine learning algorithms and experienced radiologists. By identifying the causal features that influence the model's decision, we can enforce the alignment of these causal areas with those of the radiologists through causal alignment loss. This further allows us to train hierarchical decision model that closely mirrors the expert's 's decision pipeline. The effectiveness of our approach is demonstrated by improved alignment in lung cancer and breast cancer diagnosis.
Limitation and Future works. The optimization of our causal alignment loss can be computation- ally expensive due to the estimation of the implicit Jacobian matrix. We will investigate efficient linear equation solving techniques Mou et al. (2016) address this challenge. Additionally, we plan to apply our loss to alignment learning in multi-modality models and robotic systems.
ACKNOWLEDGEMENTS
This work was supported by National Science and Technology Major Project (2022ZD01 14904) and CSFC-6247070125: SJTU Trans-med Awards Research Fund 20220102; the Fundamental Research Funds for the Central Universities YG2023QNA47; NSFC-82470106. The authors thank Xuehai Pan for helpful discussions.
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Appendix
A CAUSAL ALIGNMENT THEORY
In this section, we discuss some theoretical aspects of causal alignment. We adopt the following notational convenience. Let and Y denote the image and the predicted label, respectively. Let Yz denote the potential outcome of the predicted label under X being Let óe.c be the generation process that maps from the original image 2g and the label yo to the counterfactual image z, which relies the model parameter & and random seed Ç. We first require the following assumptions:
Assumption A.1 (Consistency). We assume that for each individual, the predicted label Y when X exactly the potential outcome Y,-
Assumption A.2. We assume 2 has unique global minimum solution.
Remark A.3. It can be shown that the global minimum of (2) can be attained via gradient descent under smoothness, and Polsai-L.oassewiizz conditions Polyak (1964); Csiba Richtárik (2017)
For deep learning optimization, the global minimum can be obtained if is is over parameterized Du
ct al. (019)) or has sufficient width Haeffele Vidal (2017) Kawaguchi & Huang (2019).
Under the above assumptions, we show the probability of causation Ps(Y, = y/ = ro. Y = yo) identifiable.
Proposition A.4. Assume Asms. A.7 A.2 then the probability of causation is identifiable with
Po(Y₂ = y|X = 2o, Y -yo) = Po = Py (xrlzo, Bo).
Proof. Denote the counterfactual generator do.c- If fix the model parameter 0 and the random seed Ç. then &e.c a deterministic function, which means the conditional probability Pe (e (20 yo) 1(x' (00 (20. yo)) =1(x' = =x) for any x'. Therefore, have
Po 5/X Y =y0) Po(Y = y/ 2o. yo) Vo)Po(r' Po(x'/wo, vo)de'
Po = y/ar, xo. yo) Pa xo, vn)-
Further, under the fixed seed ñ, the potential outcome Y, is fully determined the classifier fo and the counterfactual image x via
Y= = sign u));
where u denotes the realization of the randomness U in network prediction under the seed Ç. There- fore, we have Y, (Xo, Yo)l = x and
Po(Y, y|X Tu, Y = Mu) Po(Y, y/a, zo vo) (x Mo)
Po(Y, = yl=)Pa(<lzzo yo)
Po(Y ( 3o),
where the last equality is due to Asm. A.1 This shows the identification equation.
Below, show x* in maximizes the probability of causation, which means supp(æ* zo) represents the causal factors that determine the model's 's decisions. As result, minimizing Latign encourages the model's s causal factors to align with those of the experts.
Proposition A.S. Assume Asms.
A.7
A.2
we have
x² = arg max -
x=|== 20. x = =0o)
for some da.
Proof. We first show that 2 equivalent to the following constrained optimization problem:
arg ming (oolooo
(8)
To this end, let d,, d(x 20) and let arg min Loo (fo(z),))) we show Lcc(f((x)) vf" >d(x" mo) Lee 2do="
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Since Asm. A.2 ensures the uniqueness of the minimum of (2) it then follows that x x= and 8 holds. Now, note that x" satisfies d(xx, 20)) $ da. which means
Lee (folfo( ((xx"),0 v") Z Lee (fo()))))) Since x satisfies d()",00) S d. d(x", ro), further have
Leo (co( (,,(("),,1)) + Ad(x", vo) 2 (oo(f((")) + ad(a",ar))
Since x' minimizes we also have
Lee (oo(f(("), v²) Ad(x" xo) S Loo(fo(z'). + Ad(x" x0))
Therefore, we have (9 holds.
We then show x" maximize the probability of causation. From we have
x
arg max, #Sdd Po(Y y ( no yo):
where the term (o ro, vo) = (s 0 ( vol) represents the generating process of x, and the term PY = y (r) represents maximizing the logarithm likelihood in the cross-entropy loss.
Then, according to the identification quantity of Po - zo, Y yo) shown in Prop. A.4 we have
arg max Po '=| zo. Y yo)-
This concludes the proof.
B NETWORK ARCHITECTURES
In this section, show the network architectures used in lung nodule classification (see Fig. and breast mass classification (see Fig. ().
besign
Si-ik
Altristes Label
Figure 5: Network architecture used in lung nodule classification.
Iugg ta Alvkatn/
ASI bute Lbbel
Figure 6: Network architecture used in breast mass classification.
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C EXTRA EXPERIMENTAL RESULTS
C.I APPLICABILITY TO DIFFERENT DATA MODALITIES
In this section, we demonstrate the applicability of our method different data modalities. Specif- ically, we consider brain MRI data from the BraTTS dataset breast ultrasound data from the Aryashah2k dataset) lung CT data from the LIDC-IDRI dataset dataset/Armato Armato III et al. (201T). and breast mammogram data from the CBIS-DDSM dataset [ee et al. (2017). The results are presented in Tab. showing that our method is consistently accurate across various data types.
Table 3: Performance of our method and baselines on different data modalities. The result of our method is boldfaced and the best result among baselines underlined.
Precision of CAM
Classification Accuracy
Methodology
MRI
Ultra.
CT
Mamm.
MRI
Ultra.
CT
Mamm.
Ross el
2017
0.036
0.197
0.034
0.084
0.730
0.679
0.656
0.559
Zhang el 2018
0.168
0.159
0.068
0.110
0.698
0.764
0.381
0.581
Brendel & Bethge
2019
0.111
0.165
0.048
0.090
0.270
0.321
0.358
0,592
Rieger et al.
2020
0.097
0.184
0.041
0.232
0.099
0,509
0.343
0,586
Chang cl al
2021
0.147
0.127
0.074
0.119
0.410
0.270
0.503
0.496
Ours
0.908
0.872
0.751
0.805
0.835
0.797
0.722
0.656
C.2 RESULTS UNDER DIFFERENT SHORTCUT SYMBOLS
We then show the performance of our method under various shortcut symbol settings. Specifically, consider three cases: the +/- marker, intensity change, and the absence of symbol. The results are presented in Tab. showing that our method is effective across different shortcut symbols.
Table 4: Performance under different shortcut symbols.
Precision of CAM
Classification Accuracy
Symbol
LIDC
DDSM
LIDC
DDSM
None
0.783
0.882
0.707
0.652
Intensity
0.760
0.783
0.723
0.670
+/-
0.751
0.805
0.722
0.656
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C.3 VISUALIZATION OF COUNTERFACTUAL IMAGES
In this section, we visualize the generated counterfactual images and show the result in Fig. 7 As see, the counterfactual modifications clearly perceptible and align with specific clinical concepts, thereby validating the effectiveness of our counterfactual generation method.
Original Counterfactual
Modified
Original
Counterfactualaz
[a] Margin changes from sharp to poorly defined
(d) Texture changes from solid to GGC, spiculation from absent to present
Original Counterfactual
Original Image (x)
Counterfactual
areas (x
(b) Lobulation changes from absent to present
(e) changes fluid
Original Image (x)
Counterfactual x²
Modified amaa (x - x'1
Original image (x)
Counterfactual x"
Modified areas Ix x')
Internal structure changes from soft tissue to fluid
(1) sphericity changes from round ovoid
Figure 7: Generated counterfactual images on the LIDC-IDRI dataset. For each sub-figure, the left, middle, and right images denote the original image z, the counterfactual image x', and the modified area supp(r - )²), respectively. Positive modifications are marked red and negative marked in blue. We observe that the counterfactual modifications all correspond to certain clinical attributes of the nodule, for example, in (a), the margin attribute changes from sharp poorly defined when the label y changes from benign malignant.
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C.4 VISUALIZATION OF CAMS
In this section, we provide more visualizations of the CAMs.
Input Image
Ross et al.
ICNN
BagNets
Rieger
Chang
Ours
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Figure 8: CAM visualization on the LIDC-IDRI dataset.
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Input Image
Ross et al,
ICNN
BagNets
Rieger et al.
Chang et al.
Ours
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Figure 9: CAM visualization the CBID-DDSM dataset.
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Reviews:
[{"review_id": "i715qwo0VT", "reviewer": "Reviewer_SaCM", "summary": "This paper addresses the problem of causal alignment, aiming to recover the causal mechanisms in a decision-making process. It specifically focuses on the classification of medical images, seeking to align the model's reasoning with the causal mechanisms used by experienced radiologists. With access to radiologist-annotated areas, the authors first use counterfactual generation to identify image regions that explain the classification result and then introduce a causal alignment loss to match these identified areas with the radiologist annotations. The alignment loss is optimized using the Implicit Function Theorem and the conjugate gradient algorithm. When attribute descriptions of abnormalities are available, they select the subset of attributes that causally determined the radiologists' labeling for each patient by maximizing the Conditional Counterfactual Causal Effect (CCCE). Finally, they propose a hierarchical alignment process to align both the annotated areas and attribute descriptions with expert annotations.", "questions": "- Is the loss function in line 194 optimized directly? From Algorithm 1, it appears that this loss function is optimized in a two-step process, i.e. finding $x_i^*$ by minimizing $L_{ce}$ and then finding $\\theta$ by minimizing $L_{align}$.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "- To address the limitations in previous literature, where identified image regions align with expert behaviors only associationally, this paper proposes using counterfactual generation to identify regions that causally determine the model’s decision. The authors show that the generated counterfactual images maximize the probability of causation.\n- To align the identified causal factors with radiologist-annotated areas, the paper introduces a causal alignment loss and an algorithm to estimate the gradient of this loss.\n- The proposed method shows significant improvement over baseline methods in Class Activation Mapping (CAM) precision and classification accuracy. It effectively distinguishes spurious correlations from true causal factors.", "weaknesses": "- The selection of the set of attributes that are causally related to the label requires further justification. The paper claims that the subset $S$ maximizing the CCCE represents the causally related attributes. However, if an attribute $A_k$ is not a cause of $Y$ for some $k \\in \\{1:p\\}$, then we would have $Y_{A_k = 1, A_{-k} = a_{-k}} = Y_{A_k = 0, A_{-k} = a_{-k}}$ for any fixed $a_{-k}$. Thus, the CCCE would be the same for the true causally related subset $S$ and any superset of $S$. It seems that the full set $A$ might also maximize CCCE. However, since only six attributes are annotated in both the LIDC-IDRI and CBIS-DDSM datasets, the proposed algorithm may still work even if $r_i = (1, \\dots, 1)^T$ for all $i$.\n- The paper claims that previous methods, which regularize the model's input gradient to align with expert-annotated areas (lines 37–40), only capture associational factors. However, the paper later (lines 405–407) attributes the failure of these methods not to their inability to distinguish spurious correlations, but to the limited effectiveness of gradient-based approaches themselves. In fact, even if the identified image regions are only associational factors, regularizing the gradient to expert annotations will rule out spurious correlations. Although the explanation in Figure 1(a) is conceptually reasonable, the simulation does not explicitly demonstrate the advantage of causal factors over associational factors when expert annotations are leveraged.\n- It would be helpful to clarify Figure 3, the causal diagram that illustrates radiologists' decision process. It should be emphasized that $A_{1:p}$ represents the experts' \"decision\" about image attributes, rather than the underlying true attributes of the image. It initially appears to me that the underlying true attributes are inherent features of the image, in which case no causal relationship would exist between $X$ and these true attributes. Besides, how the area marks $m$ fit into the causal diagram could be explained further.\n- Could the authors provide the computation time for the proposed method compared to the baseline methods?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "7jf1Dzf5PA", "reviewer": "Reviewer_gNbm", "summary": "This paper proposes a causality-based alignment framework to align the model’s decision with that of experts in disease diagnosis. The key components of the framework are counterfactual generation and causal alignment loss. Counterfactual generation highlights the areas that explain the model’s decision process, and the causal alignment loss minimized the distance between expert-annotated areas/attributes and the generated counterfactual. An optimization algorithm based on Implicit Function Theorem is designed to solve this two-stage problem. The authors then adapted the framework to two scenarios: area of interest annotations and attribute annotations, and develops a hierarchical alignment structure for attribute annotations. Various experiments are conducted and the results show that the causality-based alignment framework is robust against spurious correlation.", "questions": "Please see weakness.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "This is a good paper, I think. The presentation is concise, and the explanation of theory is clear. Numerical experiments are solid, with detailed description of implementation and comparison of baselines. Also, I think the idea of designing causal alignment loss based on counterfactual generation very fascinating.", "weaknesses": "My major question or concern for the causality-based alignment framework is computation cost, or scalability. For example, when solving for $x^*$, what if there are multiple counterfactual classes, or even more challenging, $y^*$ is continuous? I guess a straightforward solution would be to bin $y^*$ into binary classes, but that might sacrifice the granularity of counterfactual generation. Also, for hierarchical alignment, conditional counterfactual causal effect score is calculated for each attribute subset. What if the attributes are high-dimensional?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "IM1pNYYr1d", "reviewer": "Reviewer_bkax", "summary": "The authors present a novel method to help researchers achieve alignment of causal attributions in image analysis for, e.g., lung disease, with that of experts. The novelty is that the alignment is grounded in causal attribution rather than mere association.", "questions": "Could you clarify the points listed in the Weaknesses section in the text and put the causal model here in the appropriate context?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "The results are compelling, and the problem it takes up is relevant and timely and promises immediate practical benefits if other researchers pursue it. Further elaboration of the idea shows promise.\n\nThe math used makes sense, given the causal model employed. I have some questions about the causal model that can be fixed in the presentation, which I describe below.\n\nI am especially impressed with Figure 4, which shows that the method can infer regions of interest that are closely aligned with those of experts, and with Table 1, which shows the overall accuracy of the method.\n\nIn short, the argument that taking causal structure into account in the problem of aligning image analysis with experts in the biomedical sciences is well-put.", "weaknesses": "The main weakness in the paper, which, as I said, can I think be addressed textually, concerns the causal diagram in Figure 3. This diagram implies that the attributes selected by the procedure are independent of each other and conditional on the existence of a mass. There is no reason I can think of that this must be the case. For example, there could be a latent variable L such that X -> L -> {A1, A2}, in which case conditioning on X would not be sufficient to separate A1 and A2. Also, there's no reason to think that relevant features cannot overlap in the image, at least not in principle, in which case they will not be independent. Some nuance needs to be added to this point. I don't believe the model needs to be changed, but I feel it needs to be presented as a heuristic choice or a choice about the types of feature sets the authors aim to discover.\n\nA second comment is that causal search algorithms are available that, with appropriate background knowledge, could also achieve a result of finding models in the form of Figure 3, and these have not been compared directly for efficacy. I will not recommend such models, as we are not supposed to in these reviews to recommend our own work. Still, the authors could search for these in the literature and explicitly state that perhaps, while they may be effective, a different approach is being taken up and evaluated here, or at least to give reasons why they are not pursued. Some of these methods are highly accurate in terms of the causal graphs that they infer, though the assumptions of these methods could be discussed briefly.\n\nA third comment is that nothing rules out, in principle, the possibility that multiple masses might occur in a single image, and it is not clear what the method would say in cases like this.\n\nA fourth comment is that what model class is being assumed here must be explicitly stated. It's a mixed linear Gaussian/discrete regime, though this should be stated, and if that's not the case, then that should especially be stated up front at the beginning of the article. If it's there and I missed it, then I apologize.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "hx9rIDxeOi", "reviewer": "Reviewer_DrbE", "summary": "This work addresses the problem of identifying medical diagnoses while also uncovering factors that indeed causally contribute to these diagnoses. The authors develop a novel learning and optimization framework that efficiently and accurately achieves this goal. Empirical results demonstrate its superior performance compared to other methods.", "questions": "Please see Weakness section.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "High Novelty/Quality: This work develops a novel causal alignment loss and provides an efficient optimization method for solving the learning problem.\n\nHigh Significance: The empirical results show that their approach not only achieves high diagnostic accuracy but, more importantly, identifies causal factors that contribute to the diagnosis. Accurate identification of causal factors is of paramount importance in the medical field, as it provides informed and precise information for planning subsequent treatments.", "weaknesses": "Clarity: \n\n1. In Figure 2 and Figure 1.b, I am confused about the terms \"calcification,\" \"margin,\" and \"spiculation.\" Are these attributes? From my understanding, which was described in the problem setup in section 3  the attributes a_i are binary indicators for the image. Do you mean that each of the binary attributes a_i, for example, indicates the presence of \"calcification\" in the image? It would be great if the authors could provide a more clear descriptions/ explanations in Figure 2 and Figure 1. b captions.\n\n2. In Equation (2) for counterfactual generation, which distance function are you using? Since the choice of the distance function will indeed affect the downstream causal factor identification and diagnostic accuracy, it would be great if the authors could provide a section for discussion or ablation study to analyze the effect of distance function on the model performance and interpretability. \n\nBesides common distance metrics (e.g., L1, L2, cosine similarity), possible prior-studied distance functions in the context of counterfactual generation are worthwhile for discussion [https://arxiv.org/pdf/1711.00399]:\n\nSquared Euclidean Distance :\n\\begin{align} d(x_i,x^c_i) = \\sum^d_{k=1} (x_{i,k} - x^c_{i,k})^2 \\end{align}\n\nScaled Squared Euclidean Distance :\n\\begin{align} d(x_i,x^c_i) = \\sum^d_{k=1} \\frac{(x_{i,k} - x^c_{i,k})^2}{std_k(x)} \\end{align}\n\nScaled L1 Norm :\n\\begin{align} d(x_i,x^c_i) = \\sum^d_{k=1} \\frac{|x_{i,k} - x^c_{i,k}|}{MAP_k} \\quad \\text{where,} \\end{align} \\begin{align} {MAP_k} = median_j (|x_{j,k} -median_l(x_{l,k})|) \\end{align}", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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:

Decision: Accept (Poster)