Quantifying Statistical Significance of Neural Network-based Image Segmentation by Selective Inference
Authors: Vo Nguyen Le Duy
Shogo Iwazaki
Ichiro Takeuchi
Venue: NeurIPS 2022 Accept (2022)
Keywords: Image Segmentation
Uncertainty Quantification
Selective Inference
Statistical Hypothesis Testing
p-value

Abstract: Although a vast body of literature relates to image segmentation methods that use deep neural networks (DNNs), less attention has been paid to assessing the statistical reliability of segmentation results. In this study, we interpret the segmentation results as hypotheses driven by DNN (called DNN-driven hypotheses) and propose a method to quantify the reliability of these hypotheses within a statistical hypothesis testing framework. To this end, we introduce a conditional selective inference (SI) framework---a new statistical inference framework for data-driven hypotheses that has recently received considerable attention---to compute exact (non-asymptotic) valid p-values for the segmentation results. To use the conditional SI framework for DNN-based segmentation, we develop a new SI algorithm based on the homotopy method, which enables us to derive the exact (non-asymptotic) sampling distribution of DNN-driven hypothesis. We conduct several experiments to demonstrate the performance of the proposed method.

Quantifying Statistical Significance of Neural Network-based Image Segmentation by Selective Inference
Vo Nguyen Le Duy Nagoya Institute of Technology and RIKEN duy.mllab. mllab.llbbb.......m nit@gmail. com
Shogo Iwazaki Nagoya Institute of Technology
iwazaki s mllab. ni nit@gmail.com com
Ichiro Takeuchi*
Nagoya University and RIKEN
ichiro. takeuchi @mae nagoya-u. ac. jp
Abstract
Although a vast body of literature relates to image segmentation methods that use deep neural networks (DNNs), less attention has been paid assessing the statistical reliability of segmentation results. In this study, we interpret the segmen- tation results as hypotheses driven by DNN (called DNN-driven hypotheses) and propose method to quantify the reliability of these hypotheses within statistical hypothesis testing framework. To this end, we introduce a conditional selective inference (SI) framework-a new statistical inference framework for data-driven hypotheses that has recently received considerable attention to compute exact asymptot))) valid p-values for the segmentation results. To use the conditional framework for DNN-based segmentation, we develop a new SI algorithm based the homotopy method, which enables to derive the exact (oon-asymptotic) sampling distribution of DNN-driven hypothesis. We conduct several experiments demonstrate the performance of the proposed method.
Introduction
Image segmentation is a fundamental task in computer vision, and it has been widely applied in many areas. The goal of image segmentation is to assign a label to every pixel in an observed image, such that pixels with the same label share certain characteristics. segmentation algorithms have been developed based on thresholding [2)) region-growing [21] or graph cuts 2 3. However, over the past few years, image segmentation using deep neural networks (DNNs) has become popular model that exhibits remarkable performance improvements [19
the literature, numerous image
Although vast body of literature relates to DNN-based segmentation methods, less attention has been paid to evaluating the statistical reliability of the segmentation results. In the absence of statistical reliability, it difficult to manage the risk of obtaining incorrect segmentation results, which might be harmful when they are used in high-stakes decision-making such as medical diagnoses or automatic driving. Therefore, it is necessary to develop a valid statistical inference method for data-driven DNN-based segmentation results that can properly control the risk of obtaining false positives.
Valid statistical inference the DNN-based segmentation results is intrinsically difficult because the observed data is used twice: once for segmentation and once again for inference. This is often referred to as double dipping IE In statistics, has been recognized that naively computing p-values in
Cooresponding author
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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double dipping is highly biased and correcting this bias is challenging, especially when the hypotheses are selected using complicated procedures such as DNN. Our idea is to introduce the conditional Selective Inference (SI) framework for resolving this challenge. The basic idea behind conditional SI to perform statistical inference conditional on the selection event (i.e., segmentation results), which allows us to derive the exact (non-asymptotic) sampling distribution of the test statistic for making the valid statistical inference.
Related works. In the past few years, conditional SI has been recognized as a new promising approach by which evaluate the statistical reliability of data-driven hypotheses. and it has been actively studied for inference on the features of linear models selected by several feature selection methods- for example, Lasso 17 13 [5] The basic idea of ST to make inferences conditional on the selection event, which allows us to derive the exact sampling distribution of the test statistic. In
addition, SI has been applied to various problems such as change point detection 32
|2
12
L
53 27].
outlier detection
5 31, generalized Lasso [ [6]. sequential feature selection 30 26 and others
28
2
10
However, no study to date provides conditional SI for DNNs.
The most closely related work (and the motivation for this study) is A where the authors provide a
framework to compute valid p-values for image segmentation results obtained by threshold (TH)- based and graph cut (GC)-based segmentation algorithms. The novel idea of [29] is to consider the sampling distribution of the test statistic conditional on the selection event, which the segmentation result is obtained. In that study, since the authors consider simple segmentation algorithms, the selection event is simply characterized by set of linear or quadratic inequalities which is represented as a single polytope in the data space. Then, they can directly use the approach in the seminal conditional SI paper 17 to conduct the inference. However, it is not the case of DNN because the selection event of the DNN is much more complicated than that of TH and GC. Therefore, the method in 129 is not applicable in the case of DNN-based segmentation.
There are several types of problem setting with various network structures for DNN- segmenta-
tion. In this study, we focus on the most standard one, the supervised segmentation setting with a simple structure (see S and $ As will be described in $ our target is to conduct the statistical inference the test phase (not in the training phase). Namely, our goal is to quantify the reliability
of the segmentation result when new test image is given to DNN that has been trained in advance.
Contributions. Our main contributions are follows:
To the best of our knowledge, this the first study to provide an exact (non-asymptotic) inference method for statistically quantifying the reliability of image segmentation results obtained from DNNs.
We propose homotopy method to conduct powerful and efficient conditional SI for DNN-based segmentation tasks. In this study, we mainly focus on standard convolutional neural network (CNN) as working example and show that all basic operations the standard CNN (e.g., convolution, max pooling) can be incorporated in our proposed homotopy- based conditional SI method.
We undertake experiments on both synthetic and real-world datasets, through which we offer evidence that our proposed method can successfully control the False Positive Rate (FPR), has good performance in terms of computational efficiency, and provides good results in practical applications.
Our implementation is available at
2
Preliminary
In this section, we first review the conditional SI method for TH-based and GC-based segmentation algorithms in (29). Then, we clarify the challenges of introducing the conditional SI for DNN-based segmentation tasks and present our idea resolve the diffieulties.
2.1 Conditional SI for TH-based and GC-based Image Segmentation
29
Consider an image with n pixels corrupted by Gaussian noise as
x (X (X,,.., X,) = tut,, E~ N(O,
(1)
2

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where E IR" is an unknown mean pixel intensity vector, and E E IR" is a vector of normally distributed noise with covariance matrix Σ, which is known or can be estimated from external data. We note that the assumption in Equation ① does not mean that the pixel intensities in an image follow a multivariate normal distribution. Instead, the vector of noise added to the true pixel values are assumed to follows multivariate normal distribution. For simplicity, we consider segmentation problems in which the image is divided into two regions: object and background regions We denote the sets of pixels in the object and background regions Ox and Bx. respectively.
Statistical inference. To quantify the statistical significance of the segmentation result, Tanizaki al. 2 consider the following statistical test:
Ho Hox = PBx vs. H1 : Mox + PBx
(2)
where fOx and M8x are the true means of the pixel values in the object and background regions. A reasonable choice of the test statistic is the difference in the average pixel values between the object and background regions
n
X
Σ X iEOx
Σ X1, sEBx
(3)
0x
1x
where r x xx
1 is the vector indicating the test-statistic direction, and 18 E R is vector whose elements belonging to set C are set to 1, and O otherwise. We would like to note that the hypotheses in 2 and the test-statistic direction r7 in 3 are data-driven because they depend on X. Given significance level E [0. 1] (c.g., 0.05). we reject the null hypothesis Ho if the p-value is smaller than a, which indicates that the object region differs from the background region. Otherwise, we cannot say that difference is significant.
Fx
8x
Conditional SI for computing valid p-values. Let us define A(X) as the event in which the result of dividing the pixels of image X into the object region Ox and the background region Bx is obtained by applying a segmentation algorithm A (i.e., TH-based or GC-based image segmentation algorithm) on X that is,
A(X) = 10x, Bx}.
(4)
To conduct valid inference based on the concept of conditional SI, P0 proposes to consider the sampling distribution of the test statistic n X conditional on the segmentation results, that is,
77 Tx | A(X) = A(x
(5)
where obs is an observation (realization) of the random image X. Then, to test the statistical significance of the segmentation result, the authors introduced the selective p-value, which satisfies the following sampling property:
PH, (Pselective I a A(X) = h(xx)))) = a,
(6)
ie., Pselective follows a uniform distribution under the null hypothesis. The Prelective is defined as
Prelective = FHo (n7 x
2 17 obs
A(X) = A(x ) x()) = q(z)
where ((X) is the sufficient statistic of the nuisance parameter that needs to be conditioned on in order to tractably conduct the inference and it is defined as (X) = (In en T) with c =
En Enn) Here, we note that the selective p-value depends on q( X). but the sampling property in 6 continues to be satisfied without this additional condition because we can marginalize over all values of (X). The ((X) corresponds to the component = in the seminal conditional SI paper [7 (see Sec. 5, Eq. 5.2 and Theorem 5.2). We note that additional conditioning q(X) is standard approach in the conditional SI literature and is used in almost all the conditional SI-related studies.
Example 1. (Selection event of TH-based segmentation and the conditional inference) Given threshold t and an observed data obs, the selection event of TH-bassa segmentation written as
A(X) b(x + Bxx 10 BBero [XER:
X; Vi gobb X,<T,
obe }
B
which is simply a single polytope in the data space. Then, the conditional inference in for the case of TT--babbed segmentation is explicitly defined
1 X (X+ T, Vi Ogota X, T, Vj B Byots }.
'The extension to the cases where the image divided into more than two regions straightforward.
3

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nalve
selective
(3) Original Image
(b) Segmentation Result
(c) Pixel Distribution
(d) Class and Class Histogram
Figure 1: Illustration of the bias in naive statistical inference. (a) We generate null image in which all the "true" pixel values are the same. (b) The segmentation result obtained by a trained CNN. (c) The distribution and the histogram of the pixel intensities. (d) The histograms of the pixel intensities in the background (Class I) and object (Class 2), as well as the naive p-value and the proposed selective p-valee. Even from image containing object, the naive p-value is very small, indicating that it cannot be used to evaluate the reliability of the segmentation result. By applying the proposed method, successfully identify the false segmentation result.
Similarly, 29 showed that the selection event of GC-based segmentation can be approximated by of quadratic inequalities which is also represented by a polytope. When the selection event is represented by a single polytope, we can directly apply the original conditional SI method by [17].
2.2 Challenges of Introducing Conditional SI for DNN-based Image Segmentation
In this paper, consider the same problem setup as described in 29 (i.e., from to Z)). The main difference that the segmentation event A = {Ox Bx in is now obtained from a DNN that has already been trained in advance for the segmentation task. The discussion thus far indicates that we can conduct an exact statistical inference for the segmentation result if we can compute the selective p-value in 7 However, to compute Pselective, the major challenge is characterize selection event (A(X) A(xxbs)) }} of DNN-based segmentation, which is much more complicated and cannot be simply represented by intersection of linear quadratic inequalities in the of TH-based and GC-based segmentations. In the next section, we propose a novel method that addresses all the aforementioned challenges by utilizing the concept of parametrized line search (i.e., the homotopy method). Figure iillstrrates the selection bias in naive statistical inference and the importance of the proposed method.
3 Proposed Method
In this section, we propose a method for computing the selective p-values in 7 The main task is to identify the following set of a R. that satisfies the condition part:
X'
Rd A(x))
q(x) bbs
o
q( obs)
To identify X, we first reformulate the problem of identifying this data space as problem of searching the data on a parametrized line. Then, introduce homotopy method to conduct a parametrized line search illustrated in Fig. 2
3.1 Characterization of the Conditional Data Space X'
In according the second condition on the sufficient statistic of the nuisance parameter qxx), the data X' is restricted to a line R", as stated in the following lemma.
Lemma 1. Let us define
obe) and b Em(nTT)) Emn
(9)
Then, the conditional data space can be rewritten using a scalar parameter E IR as
(= = a bz E 2}, where = {a E A(x(z)) = A(zz
(10)
Proof. The proof is deferred to Appendix A

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Observed
Segmentation Result
DNN (x ohe)
onlix pio
60% spal
dfii Sp
Parametrized
x-
X62
Proposed Method
Conditional Data Space
Data Space R
[x
R
(d(x) a'x tbs), q(x) ((x))
A schematic illustration of the proposed method. By applying a trained CNN to the observed image xhb we obtain a segmentation result. Then, we parametrize xb with a scalar parameter in the dimension of the test statistic to identify the subspace X' whose data have the same segmentation result as abs Finally, the valid statistical inference is conducted conditional on X. We introduce a homotopy method for efficiently characterizing the conditional data space X'
Figure 2:
We note that whereas the seminal conditional work 17 already implicitly considers the data on the line, this was explicitly discussed for the first time in Section 6 of [8. Lemma I lindicates that we need not consider the n-dimensional data space. Instead, we need only consider the one-dimensional projected data space 2 in (T). Now, let consider random variable Z IR and its observation oob E R that satisfies X- a bZ and abs
bzobe, The selective p-value is rewritten as
HHo (1n T X 2 |n
x obs
X E X)
Pselective
PHo (IZ) obs Z
Z
(11)
Because the variable Z N(0, 1) En) under the null hypothesis, Z Z Z follows truncated normal distribution. Once the truncation region Z is identified, computation of the selective p-value in is straightforward. Therefore, the remaining task identify Z.
3.2 Identification of Truncation Region 2
As discussed in 3.1 to calculate the selective p-value we must identify the truncation region 2 in (TO). To construct Z, we must (a) compute A(xx(z) for all E IR, and (b) identify the set of intervals of z which A(æ(2)) A(zobs) However, it seems intractable to obtain A(x(2)) for infinitely many values of z E IR.
Our first idea of introducing conditional SI for DNN is that we additionally condition on some extra events make the problem tractable. We now focus on a class of DNNs whose activation for example. ReLU, Leaky ReLU, Then, we consider additional conditioning on the selected piece each piecewise-linear AF.
functions (AFs) are piecewise-linge
Definition 1. Let 8, (z) be "the selected piece" of piecewise- AF the j-th unit in a DNN for a given input image x, and let s(x) be the set of 8j (x) for all the nodes in a DNN.
For example, given ReLU AF, s5(x) takes either 0 depending on whether the input the j-th unit located at the flat part (inactive) or the linear part (active) of the ReLU function. Using the notion of selected pieces s(x), instead of computing the selective p- p-value in IT.. we consider the following over-conditioning (oc) conditional p-value:
oc P Prelective = PH, (lZ) Z 1 |b
1 ZOO),
(12)
where
DC f= E A(x(=)) = A(x s<x(()) = 8 (cc oha
However, such over-conditioning in SI leads loss of statistical power 1n
Our second idea to develop a homotopy method to resolve the over-conditioning problem- that is, remove the conditioning of s(x(=))= s( bbz) Using the homotopy method, we can efficiently compute A(r(z)) in a finite number operations without the need to consider infinitely many values
of z IR, which is subsequently used to obtain the truncation region Z in (10) The main idea is to
5

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case
max posting
uppample
cont thrsshodiigg
max pool
Ingu-
CWP
upssmple
Dirput Segmantation Result
R Ky
Ky
+0
Selection Event (linear inequalitin))
(a) The over-conditioning selection event of basic CNN. (b) Basic network structure for segmentation.
Figure 3: (a) The over-conditioning selection event of a basic CNN can be written as set of linear inequalities because the basic operations (e.g., convolution, max pooling) and activation functions (e.g., ReLU, thresholding) of a CNN can be represented as linear inequalities. Note that non non-piecewis, linear functions such as sigmoid or hyperbolic tangent function are equivalent to consider the values prior to activation with threshold value O. (b) Basic network structure for segmentation.
compute a finite number of breakpoints at which one node of the network changes its status from active to inactive vice versa. This concept is similar to the regularization path of Lasso, where can compute a finite number of breakpoints at which the active set changes.
To this end, we introduce two-step iterative approach, generally described as follows (see Fig.
Step (over-conditioning step). Considering the over-conditioning case by additionally condi- tioning on the selected pieces of all hidden nodes in the DNN.
Step (homotopy step). Combining multiple over-conditioning cases using the homotopy method to obtain A(x(2)) for all z E R.
3.3 Step 1: Cver-Condi Step
We show that by additionally conditioning on the selected pieces s ((xhh)) of all the hidden nodes, the selection event of DNN is written as a set of linear inequalities. The illustration shown in Fig. Ba
Lemma 2. Consider a class of DNN that consists of affine operations and piecewise-linear AFs. Then, the overconditioning region is written as
xx2) S (5(a(zobe))
go
{2 E R
using matrix (((x=)))) and vector uf (a((a))))) that depend only the selected pieces s(x
Proof. For the class of DNN that consists of affine operations and piecewise-lineal AFs, by fixing the selected pieces of all the piecewise-lineal AFs, the input each AF is represented by affine function of an image æ. Therefore, the condition for selecting a piece in a piecewise-linear AF sz(x(2)) 8y obbs),, written linear inequality with respect x(z).
Remark 1. (Activation functions in the hidden layers) In this work, focus trained DNN where the AFs used at hidden layers are piecewise linear-for example, RELU, Leaky ReLU, which commonly used in CNN. Otherwise, if there is any specific demand to use pinceeweiese linear functions (such as sigmoid or hyperbolic tangent), we can apply a piecewise- approach to these functions. We provide examples of such approximation in Appendix D
approximation
Remark 2. (Operations in trained neural network) Most basic operations in trained neural network are written as affine operations. a traditional neural network, the multiplication results between the weight matrix and the output of the previous layer and its summation with the bias vector is an affine operation. a CNN, the convolution and upsampling operations are obviously affine operations. Even when operation in DNN is NOT represented that the operation is represented by as sei of affine operations, as in the following remark.
single affine operation, it is often the case
Remark 3. (Max-pooling) Although the max pooling is not an affine operation, it can be written as set of linear inequalities. For instance, v1 = max(21, 02 Vy} can be written as a set {el v S e, V. e, v & eg v). where a = (#1, 02 03) and e; is the standard basis vector with 1 at position i. Remark 4. (Activation functions at the output layer) In Remark D we mention that we need to perform piccewise linear approximation for non-piecewisc linear activations. However, if these

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Algorithm parametric_st.
Algorithm 2 compute_ campute_sooutton_Pee
Input: ab Amia. Smax
Input: a, Amin: Smax
1: Obtain A(w applying the trained DNN abo
1:
1,
Fmin: (#+)
2: Compute 1) + Eq B
2: while 2, < Emax do
3: Calculate a and b Eq.
3: Compute an(2r)
+ bas in R" of #e
4: A(x(=)) + compute_solutionn solution path(a, b, Smin Smax) #f Algo-
4: Obtain A(n((=))) )) by applying trained DNN ar(x))
rithm
5: Compute Rx+1 e- Lemma B
5: Identify 2 + (a A(z((=)) = A(w)))))
6: T U (#111) and
6: Paclective Eq.
7: end while
Output: Psolective
Output: (A(a())))=ET
functions are used at the output layer (e.g., sigmoid function or tanh function), we need not perform the approximation task, because can define the set of linear inequalities based the values prior to activation. See the next Example for the case of a sigmoid function.
The list of components that are commonly used in DNN for segmentation task and can be represented We would like note that, since we are
as a set of linear inequalities is provided in Appendix E primarily interested in the reliability of a trained network given new inputs (i.e., not training inputs), the validity our proposed method does not depend on DNN training. Therefore, all of the operations that only operate in the training process such as Dropout do not affect the applicability of the proposed SI approach and we do not need consider the selection event of those operations our method.
(The selection event of basic CNN is a set of linear inequalities) Consider a basic network structure for segmentation as shown in Fig. 3b and n is an even number. us start with the first convolutional layer with 4 filters. Let R Rvaxy'n be the matrix obtained by the f filter. We apply ReLU activation on R/, vʃ E [4]. The selection event of ReLU is written as
Example 2.
VT V SRELU= = 000
z 0, if output of ReLU 2 O,
0, otherwise.
f=1 ==1==1j=1
Let pf E RV is the matrix after applying ReLU on R We now move to the pooling layer by applying the max max-pooling on Pl, Vf E [4], with 2 x 2 window. Selection event of max max-pooling
OUUE f=liEZ JEI
Pl Sa pf 10+1) S Sq. (111)) Sa. P P((1)))) S sa).
SMaxPool
pf 10+1) (i+1))* (0+00+)) and I 12x +1+ : (0,1, (n/2 1)}}. We continue the forward process until the last convolutional layer. Let W IR RVáxvv be the matrix at this final layer. We apply sigmoid function on W. The selection event of sigmoid is written as
where q max P!
p!
TT Vπ UU =1j==
Wi 2 0, if sigmoid output 2 0.5,
SSigmoid
W < 0, otherwise.
Finally, the over-conditioning selection event
S= SReUU U
SMaxPool
U
Ssigmoid,
which is set of linear inequalities.
3.4 Step 2: Homotopy Step
We introduce homotopy method to compute A(x(2)) by combining multiple over-conditioning steps. Lemma 3. Consider a real value zt. By applying a trained DNN to a12z), we obtain set of linear inequalities ((e(a(()))) S (/(a(=(())) Then, the next breakpoint 2+11 > 2t, at which the status of one node is changed from active to inactive or vice versa versa-thal that is, the sign of one linear inequality changes. This breakpoint calculated by
(a(a(z)))) ((()((())))
2t41
min (1(g(tt(=()))))0)
(O(o(e(z)))))
7

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1.0
1.0
propreed reeledd
1.0
proposed method
1.-
mopoeddmmhhed
proposed mutiodicc
parmutatoo
0.0
naNe
0.6
0.2
00
16
04
255
1024
0.0
16
04
256
1004
0.0
0.5
1.0
20
0.
05
n.:m
dota
(a) Independence
(b) Correlation
(a) Independence
(h) Correlation
Figure 4: False Positive Rate (FPR) comparison. Figure 5: True Positive Rate (TPR) comparison.
The proof of Lemma 3 is provided in Appendix B Algorithm
2 shows our solution to efficiently 22 < 217) are computed one by one. Each breakpoint 2,, t E [[T]] indicates the point at which the sign of one linear inequality is changed, i.e., the status of one node in the network will change from active to inactive vice versa. By identifying all these breakpoints ]E(TT]) the solution path is given by A(æ(=)) = A(x(zz)) if
identify A(x(z)). In this algorithm, multiple breakpoints 21
2 [24, 24+1, t E [IT]]. Algorithm I shows the entire procedure to calculate the selective p-value in First, we apply DNN on a: to obtain the segmentation result A(x Next, we calculate n
which is used to identify parameterized line in R". Then, we compute A(x(22) by using Algorithm
2 After obtaining Azx(z)), we identify Z in (10). which the key factor for computing the selective v-vulue. Regarding the choice of [zmin: Rmax] under normal distribution, neither very positive nor very negative z values affect the inference. Therefore, it reasonable to consider a range of values, e.g.. 20,, 20o]
18 where a is the standard deviation of the distribution of the test statistic.
Complexity. In the literature of homotopy method (a.k.a. parametric programming), it is known that the actual computational cost differs significantly from the worst case. well-known application of the homotopy method in the ML community is the Lasso regularization path, which also has the worst-case computational cost on the exponential order of the number of features, but the actual cost is known to be nearly linear order. Similarly, in the proposed homotopy method for SI, it is also evident from our experimental results that the number of breakpoints almost linearly increasing in practice (Fig. 6. We additionally note that, at step 4 of Algorithm 2 we apply trained DNN to the parametrized data a=x)) at any breakpoints 24. Therefore, the number of forward passes of the network is equal to the number of breakpoints.
Class of networks. The proposed method can be applied to a class of piecewise linear networks whose network operations are characterized by set of linear inequalities approximated by piecewise-linear functions (many state-of-the-art image segmentation networks are or can be well- approximated by piecewise linear networks). This is due the reason that all the theories and algorithms in S only depend on the property of each component and does not depend on the entire structure of the network. We believe that our method fairly general because most of the basic operations in a trained neural network can be decomposed or approximated by affine operations as we presented in Remarks Example as well as Appendix D
Experiment
Experimental setup. We compared the proposed method (homotopy method) with proposed-metiee method- oc (over-condee conditiooii. naive method and permutation test. The details of the methods for comparison are shown Appendix F
We considered two covariance matrices:
Σ I, (independence) and
. Σ (0.5/i-11/,3 E "x (correlation).
Regarding the FPR experiments. we generated 120 null images X
NO... Σ) for
n {16, 64, 256, 1024]. For each value of n, we run 120 trials. To test the power, we generated images X with = 256, in which the true average difference in the underlying model μo0, PB Ap {0.5, 1.0, 1.5, 2.0]. For each A,. we run 120 trials. We selected the significance level
0.05 and we used the basic CNN shown in Fig. 3b
8

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800
encounted intervals
400
(20 derbitonn
jeetimated dyma)
200
aptaned6
0.4
1024
Figure 6: The number of encountered intervals on the line along the direction of the statistic almost linearly increasing in practice.
38
255
1004
20
54
255
2004
Figure 7: Robustness of the proposed method.
Input Image
Segmentation
Image
Segmentation
Input Image
Segmentation
(a) Puaive 0.00, Psolective 0.67
(b) Poaive 0.00, Pselective 0.45
(c) Pnaive 0.00, Penlective 0.16
Figure 8: Inference segmentation results for images without tumor region.
Input Image
Segmentation Result
Input Image
Segmentation Result
Input Image
Segmentation Result
(a) Poaive 0.00, Prelective -8.15--5 (b) Puaive 0.00, Pselective 9.2e-3 (c) Puaive = 0.00, Pselective 1.91e-2
Figure Inference on segmentation results for images where there exists a tumor region.
In the literature, many network architectures have been proposed for segmentation problems. Al- though these architectures are very complex, if they are broken down into smaller elements, each can be represented (or least well approximated) as affine operation whose selection event can be handled by the proposed method. Because the main contribution of this study is to introduce condi- tional SI by which to evaluate the reliability of the segmentation results, empirical implementations and evaluations of the complex architectures are beyond the scope of this study.
Numerical results. The results of the FPR control shown in Fig. The proposed methods successfully control the FPR in both cases of independence and correlation while the naive and permutation methods cannot. Because the naive and permutation methods fail control the FPR, we no longer consider the power. Fig. 5 shows that the homotopy method has higher TPR than the over-conditioning option. Fig. shows the reason why the proposed homotopy method efficient. Intuitively, to overcome the over-conditioning issue, we need to consider all combinations of the activenesses of the nodes in a trained DNN, which is exponentially increasing. With the homotopy method, we need only consider the number of encountered intervals on the line along the direction of the test statistic, which almost linearly increasing in practice. Additionally, we confirm the robustness of the proposed method in terms of FPR control by applying our method to data following Laplace distribution, skew normal distribution (skewness coefficient 10), and 22 distribution. We also consider the case where variance is also estimated from the data. We test the FPR for both 0.05 and a 0.1. The FPR results are shown in Fig. Our method still maintains good performance. Regarding the last plot (estimated sigma) in Fig. since want to empirical check the performance

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Table 1: False positive rate and power comparisons in brain image dataset.
FPR
Power
Proposed Method
0.057
0.683
Permutation Test 0.640 (unreliable)
N.A
on FPR control when breaking the assumption of Σ (i.e., E is known or estimated from independent data), set = s²1 where s² is the empirical (sample) variance and is the identity matrix.
Real-data examples. We examine the brain image dataset extracted from the dataset used in 4 which includes 939 images with tumors and 941 images without tumors. We first compare our method to a permutation test terms of FPR control. The results are shown in Table I Because the permutation test cannot properly control the FPR, comparison of power is no longer needed. Comparisons of the naive p-value and the selective p-value are shown in Fig. [ and Fig. 2 The naive p-value remains small, even when the image has no tumor region; this indicated that naive p-values cannot be used to quantify the reliability of DNN-hased segmentation results. The proposed method successfully identifies false positive and true positive detections.
Discussion
We propose a novel conditional SI method to conduct inference on the DNN-based image segmenta- tion result. We believe that this study stands significant step toward reliable artificial intelligence and opens several directions for statistically evaluating the reliability of DNN-driven hypotheses. Some open questions remain, follows:
The proposed method currently does not support softmax and batch normalization operators. Although these operators can in theory be approximated by affine operations, doing so will be challenging in reality, as we need to work in multidimensional space. Although it is difficult to apply the proposed method to network architectures containing components that are difficult to decompose into affine operations, one possible solution is to devise a new network architecture that can be expressed set of affine functions but still has state-of-the-ee segmentation performance.
Although there no technical limitation in applying our proposed method state-of-the-an -fethh-rrt network structures, it requires a considerable amount of implementation effort. This is because, for large and complex network, we have implement the selection events for all the components in the network. The future direction could be first to implement the proposed method for few commonly used network structures, and then develop a generic software tool that can automatically compute the selection event when the network structure is given.
Even though this study mainly focuses on the image segmentation task, the proposed method can be applied to a class of problems where (1) the test statistic is defined as a linear contrast w.r.t the of linear
data and (2) the operations of the trained neural network can be characterized by inequalities (or approximated by piecewise-lines functions). Therefore, widening the applicability of the proposed method to other computer vision tasks as well as to other fields such as natural language processing and signal processing-woot also stand as a valuable future contribution.
Acknowledgments and Disclosure of Funding
We thank the anonymous reviewers and chair for their comments. This work was par- tially supported by MEXT KAKENHI (20H00601), IST CREST (JPMJCR2ID3), JST Moonshot R&D (JPMIMS2033-05), JST AIP Accelcration Research (JPMICR2IU2) NEDO (JPNP18002. JPNP20006), RIKEN Center for Advanced Intelligence Project, and RIKEN Junior Research Asso- ciate Program.
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References
[1] V. Badrinarayanan. A. Kendall, and R. Cipolla. Segnet: deep convolutional encoder-decoder architecture image segmentation. IEEE Iransactions on pattern analysis and machine intelligence, 39(2))2481 2495, 2017.
[2] Y. Boykov and G. Funka-Lea. Graph cuts and efficient nd image segmentation. International journal of computer vision, 70(2):109-133 2006.
[3] Y. Boykov and M.-P. Jolly. Interactive graph cuts for optimal boundary & region segmentation of objects in nd images. In Proceedings eighth IEEE international conference on computer vision. ICCV 2001, volume 1, pages 105-112. IEEE, 2001.
[4] M. Buda, A. Saha, and M. A. Mazurowski. Association of genomic subtypes of lower-grade gliomas with shape features automatically extracted by deep learning algorithm. Computers in biology and medicine, 109:218-222 2019.
[5] S. Chen and J. Bien. Valid inference corrected for outlier removal. Journal of Computational and Graphical Staristics, pages 1-12, 2019.
[6] Y. T. Chen and D. M. Witten. Selective inference for k-means clustering, arXiv preprint arXiv:2203.0m 152677 2022.
[7] D. Das, L. Duy, H. Hanada, K. Tsuda, and Takeuchi. Fast and more powerful selective inference for sparse high-order interaction model. arXiv preprint arXiv:2106.0493 2021.
V. N. L. Duy, H. Toda, Sugiyama, and I. Takeuchi. Computing valid p-value for optimal changepoint by selective inference using dynamic programming. Advances in Neural Information Processing Systems, pages 11356-1 1367, 2020.
[9] W. Fithian, D. Sun, and J. Taylor. Optimal inference after model selection. arXiv preprint arXiv:1410.2597. 2014.
[10] L. L. Gao, J. Bien, and D. Witten Selective inference for hierarchical clustering, Journal of the American Statistical Association, pages 1-11, 2022.
[11] S. Hyun, M. G'se G'sell, and R. J. Tibshirani. Exact post-selection inference for the generalized lasso path. Electronic Journal of Statistics, 1053:1003-1097 2018.
[12] S. Hyun, K. Lin, M. G'Sell, and R. J. Tibshirani. Post-selection inference for changepoint detection algorithms with application copy number variation data. arXiv preprint 1r11v:1812.03644, 2018.
[13] S. Jewell, P. Feamhead, and D. Witten. Testing for change in mean after changepoint detection. arXiv preprint arXiv:1910004291, 2019.
[14] N. Kriegeskorte, W. K. Simmons. S. Bellgowan, and C. I. Baker. Circular analysis in systems neuro- science: the dangers of double dipping. Nature neuroscience, 12(5):335-540, 2009.
[15] N.N. Le Duy and I. Takeuchi. Parametric programming approach for more powerful and general lasso selective inference. International Conference on Artificial Intelligence and Statistics, pages 901-909. PMLR, 2021.
[16] V. Duy and Takeuchi. More powerful conditional selective inference for generalized lasso by parametric programming. Journal of Machine Learning Research, 1330 2022.
[17] J. D. Lec, D. L. Sun, Sun, and J. E. Taylor. Exact post-selection inference, with application to the lasso. The Annals of Statistics, 44(3)9007-977, 2016.
[18] K. Liu, J. Markovic, and R. Tibshirani. More powerful post-selection inference, with application to the lasso. arXiv preprint arXiv: /801 09037, 2018
[19] J. Long, E. Shelhamer, and T. Darrell. Fully convolutional networks for semantic segmentation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1431-3448 2015.
[20] A. C. Neufeld, L. L. Gao, and D. M. Witten. Tree-values: selective inference for regression trees. Journal of Machine Learning Research, 111305):1-3.. 2022.
[21] R. Nock and F. Nielsen. Statistical region merging. IEEE Transactions on pattern analysis and machine intelligence, 26(11):1452 1458, 2004.

---

[22] N. Otsu. A threshold selection method from gray-level histograms. IEEE transactions systems, man, and cybernetics, 91):62-66, 1979.
[23] S. Panigrahi, S. Mohammed, A. Rao, and V. Baladandayuthapani. Integrative bayesian models using post-selective inference: case study in radiogenomics. arXiv preprini 22ti::2000..20122.. 2020.
[24] S. Panigrahi and J. Taylor. Approximate selective inference via maximum likelihood. Journal of the American Statistical Association, (just-aceepted)11 22, 2022.
[25] O. Ronneberger, P. Fischer. and T. Brox. U-net: Convolutional networks for biomedical image segmentation. In International Conference on Medical image computing and computer-axsisted intervention, pages 234- 241. Springer, 2015.
[26] K. Sugiyama, V. N. L. Duy, and Takeuchi. More powerful and general selective inference for stepwise feature selection using the homotopy continuation approach. In Proceedings of the 38th International Conference on Machine Learning, 2021.
[27] R. Sugiyama, H. Toda, V. N. Duy, Y. Inatsu, and Takeuchi. Valid and exact statistical inference for multi-dimensional multiple change-points by selective inference. arXiv preprint arXiv:2110.. 10.08989, 2021.
[28] S. Suzumura, K. Nakagawa, Y. Umezu, K. Tsuda, and I. Takeuchi. Selective inference for sparse high-order interaction models. In Proceedings of the 34th International Conference on Machine Learning- Volume 70, pages $338-3347 JMLR. org, 2017.
[29] K. Tanizaki, N. Hashimoto, Y. Inatsu, H. Hontami, and I. Takeuchi. Computing valid p-values for image segmentation by selective inference. In Proceedings of the IEEE/CVF Conference on Computer Vision and Puttern Recognition, pages 9553-9566 2020.
[30] R. J. Tibshirani, J. Taylor, R. Lockhart, and R. Tibshirani. Exact post-selection inference for sequential regression procedures. Journal of the American Statistical Association, 1 111(514):600 620, 2016.
[31] T. Tsukurimichi, Y. Inatsu, V. N. L. Duy, and I. Takeuchi. Conditional selective inference for robust regres- sion and outlier detection using piecewise- homotopy continuation. arXiv preprint 1:Xiv:2104.0884 2021.
[32] Y. Umezu and I. Takeuchi. Selective inference for change point detection in multi-dimmnsional sequences. arXiv preprint 1:Xiv:7066000 14, 2017.
[33] R. Valiollahi Mehrizi. Change point analysis in piccewise polynomial signals using trend filtering. 2021.
Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
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(b) Did you describe the limitations of your work? [Yes] Please see Sec. 5
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Reviews:
[{"review_id": "Y5kKv13nTGj", "reviewer": "Reviewer_W9tv", "summary": "This paper concentrates on evaluating the statistical reliability of the segmentation, necessary to develop a plausible statistical inference methodologies. This is difficult in a DNN context. Addressing such a challenge allows to derive exact (non-asymptotic) sampling distribution of the test statistic for obtaining valid statistical inference.\nThe main goal in this paper t is to conduct the statistical inference in the test phase, specifically, the goal is to quantify the reliability of the segmentation result when a new test image is given to a DNN.\n", "questions": "I consider that this paper is well written, with a clear exposition of the ideas, as well as its analytical development. However, I could no follow all the details provided, as I describe next:\n- Perhaps I did not follow the eq.(10) - left side -> x(z) = a+ b*z, I performed some calculations and I obtained  x(z) = a+ b*eta^T * z. \nSo I wonder if it is possible to obtain this derivation ? \n\n-Seems that in the process of segmentation  we obtain a large number of breakpoints, since the breakpoints are dependent on the number of the neurons in the architecture. Is this a disadvantage of the proposal ? \n-By the way, is it possible to have an idea of the  order of magnitude concerning the number of changes in a given segmentation task ?\n-The equation in Lemma 3, is computed for each neuron ? If so, I think this is a limitation of the framework\n-I understand the authors are concerned to “close” all the operations in a DNN, concerning network operations, activation and pooling functions. However, and following the material provided in the paper, I find a bit short the justifications in Remarks given (page 6). For instance, the universal polling does not fit in Remark 3, that must be, somehow, extended.\n \n(see Junhyuk Hyun, Hongje Seong, Euntai Kim, Universal Pooling -- A New Pooling Method for Convolutional Neural)\n\n- Finally, Although I think that this can constitute an advance in terms of refining the p-value, and consequently, the segmentation confidence, these contributions seem a bit short, when evaluating more complex deep network architectures.  So I may guess that, perhaps this theoretical framework may not be extended to a more drastic situation, where the comçlxity is greatly enlarged. \n\n", "limitations": "See my questions  above. But fundamentally, extending this framework to more complex architectures is not straightforward. ", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\nSelective Inference (SI) is known in the community and has been applied in a quite diverse range of applications. However, delving into details in DNN context is a less explored topic. Furthermore, proposed methodologies such as [24] are not applicable in the case of DNN-based segmentation, since the selection event is difficult to be accomplished in this context, as such this  constitutes strength of the proposed approach.\nI also think that the background and the theoretical preliminary ingredients of the SI are well written in Section 2.\n \nWeaknesses \nAlthough I think this is a valuable contribution to this research field, I have some questions that perhaps constitute the main weaknesses of the present paper (see my questions below).\n", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "5YSq0MYLpKu", "reviewer": "Reviewer_BVA9", "summary": "The paper proposes a method that extends previous work on conditional selective inference (SI) for statistical significance quantification in segmentation models to deep neural networks with affine operations and piecewise linear activation functions (the previous method was applicable to threshold- and graph-cut-based algorithms only). The core idea of this approach is to correct for the segmentation bias and perform SI on a conditional subspace of images that yield the same segmentation result as that actually observed. To identify this subspace, the paper proposes to perform search on a line parametrized by a scalar $z$ and reduces the core problem to finding the set of $z$'s compatible with the actual image (\"truncation region\"). To do this, it proposes to overcondition on \"selected pieces\" of the activations of the network nodes, shows that this can be expressed in the form of linear inequalities, and that multiple overconditioning cases can be combined with a homotopy method by finding \"breakpoints\" where exactly one network node changes its value. The method is experimentally evaluated on synthetic random data and on real images of brain tumors, using a simple conv-pooling network.\n", "questions": "- The paper generally presents the method as \"efficient\" in a few places, and on page 8 mentions that the homotopy method can be exponential in the worst case, but tends to be linear in practice. I think it would be helpful for practitioners to also mention explicitly in the text that the linear complexity has a high \"constant factor\" in the form of neural network inference (obvious in Alg. 2, but a bit hard to see from just skimming the paper).\n- nit: \"Therefore, all of the operations that only operate in the training process such as Dropout or Batch Normalization\"; this is not strictly true about dropout and BN, as these operations also affect inference (albeit in a way that is compatible with the claims in the paper and does break the assumption of the operations being affine)\n- in ref [24], it is assumed that $\\Sigma$ can be estimated from a null image known not to contain any \"foreground\" pixels; It would be useful to mention how it was estimated to generate the results in Fig. 7.\n- nit: in Alg. 2, the initialization of T should be {z_t} instead of $z_t$\n", "limitations": "No concerns.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper is very well written, cites the relevant literature, and includes a brief introduction to SI for simple segmentation methods, summarizing the results from ref [24]. The reduction of the data subspace to 1D is based on prior results from ref [16], which the present paper acknowledges. The major contribution of the present work is showing how overconditioning and the homotopy method allow the application of SI to neural network-based segmentation models, and the demonstration of its effectiveness on sample synthetic and real data. This is a novel result and should be of significant interest.\n\nThe paper is generally upfront about the limitations (affine operations only, hard to approximate softmax), but could be a bit more clear about the computational complexity, which is important for real-life applications. \n\nIt is also commendable that the authors included the code necessary to reproduce all the experiments and the figures.", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "2br2DvZjNaK", "reviewer": "Reviewer_6Pq7", "summary": "This paper studies the statistical reliability of segmentation results obtained by Neural Network-based Image Segmentation. The authors developed a conditional selective inference (SI) framework to compute exact valid p-values for the segmentation results. The experimental results on the brain image dataset demonstrate the effectiveness of the proposed method.", "questions": "Please refer to the Question in the Weaknesses.", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "[Strengths]\n+ The studied problem is interesting and important.\n+ Compared to the Permutation Test, the proposed methods could bring significant improvements.\n\n[Weaknesses]\n- The authors just conduct the experiments on a small dataset, which makes it difficult to evaluate the robustness of the proposed method. \n- The evaluated NNs model is somehow simple.  There are many SOTA image segmentation / Semantic segmentation methods, e.g., Deeplabv3+, and datasets, e.g., Pascal VOC. Could you evaluate these segmentation methods on the dataset with the proposed method?\n- What’s the relationship between Uncertainty Estimation for Semantic Segmentation and statistical reliability?\n", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept