Longitudinal Enrichment of Imaging Biomarker Representations for Improved Alzheimer's Disease Diagnosis
Authors: Saad Elbeleidy
Lyujian Lu
L. Zoe Baker
Hua Wang
Feiping Nie
Venue:  (2020)

Abstract: Longitudinal data is often available inconsistently across individuals resulting in ignoring of additionally available data. Alzheimer's Disease (AD) is a progressive disease that affects over 5 million patients in the US alone, and is the 6th leading cause of death. Early detection of AD can significantly improve or extend a patient's life so it is critical to use all available information about patients. 
We propose an unsupervised method to learn a consistent representation by utilizing inconsistent data through minimizing the ratio of $p$-Order Principal Components Analysis (PCA) and Locality Preserving Projections (LPP). Our method's representation can outperform the use of consistent data alone and does not require the use of complex tensor-specific approaches. We run experiments on patient data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI), which consists of inconsistent data, to predict patients' diagnosis.

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LONGITUDINAL ENRICHMENT OF IMAGING BIOMARKER REPRESENTATIONS FOR IMPROVED ALZHEIMER'S S DISEASE DIAGNOSIS
Anonymous authors Paper under double-blind review
ABSTRACT
Longitudinal data is often available inconsistently across individuals resulting in ignoring of additionally available data. Alzheimer's Disease (AD) is progres- sive disease that affects over 5 million patients in the US alone, and is the 6th leading cause of death. Early detection of AD can significantly improve or extend a patient's life
critical use all available information about patients.
We propose an unsupervised method to learn a consistent representation by utiliz- ing inconsistent data through minimizing the ratio of p-Order Principal Compo- nents Analysis (PCA) and Locality Preserving Projections (LPP). Our method's s representation can outperform the use of consistent data alone and does not require the use of complex tensor approaches. We run experiments on patient data from the Alzheimer's ' Disease Neuroimaging Initiative (ADNI), which consists of inconsistent data, to predict patients' diagnosis. Our results show that we can uti- lize additional data about patients, even when they are inconsistent, to improve their diagnosis accuracy thereby improving or lengthening their lifetime. We also identify key biomarkers predictive of AD consistent with clinical research.
INTRODUCTION
The use of longitudinal data, or data of the same nature collected over time, in machine learning a heavily studied field. Medical data traditionally comes in this format and so machine learning applications longitudinal data are inherently effective in patient disease diagnosis.
Alzheimers Disease (AD) affects millions Americans each year and its seriousness is evident with it being the 6th leading cause of death in the US, (Association al. 2018). As time passes, AD patients' memory and mental abilities decline. Medical research in (Logsdon et al.) 2007 Leifer 2003) has shown that the early detection of AD can vastly improve patients' lives and potentially prevent the disease becoming deadly. Since AD's progression is important to track, subjects generally conduct multiple medical scans monitor the disease's 's progression. Two general problems arise with longitudinal data; complexity of tensor methods and inconsistency of data over time.
If we consider each patient to have a vector of features at single time point, the patient is repre- sented as matrix when viewed over time and set of patients as tensor. Representing the problem as a tensor can cause difficulty when predicting the diagnosis of a patient due to the missing data. This problem heavily studied with (Wang et al. 2012: Brand et al. 2018, Lu et al., 2018)) as example approaches which, despite their effectiveness, present an added complexity to the problem.
As the number of samples and time-steps increases, we find inconsistency across the data since samples will have some missing data at varying time points. This is exceedingly common in medical data due to the difficulty and cost of ensuring all patients conduct scans at the same time. Many approaches can be applied to handle missing data, however researchers most commonly select the subset of patients who have consistently available data. This approach discards a substantial amount of available data that could be used to provide value in predicting patients' diagnosis.
Our approach is to learn a new representation for each patient that incorporates all their available data, while having consistent dimensions with other patients. This consistent representation allows

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us apply classical supervised learning methods to problem and can be learned from inconsis- tent longitudinal data. In our chosen application, we focus AD diagnosis from MRI scans where all subjects received an initial scan that refer to the baseline. Subjects returned to receive more scans at varying time-steps such as 6, 12, and 24 months later but not all subjects conducted a sean all the time points. We enrich the baseline representation using additional scans of each patient by balancing between the global patterns and pairwise local patterns across the patient's 's data.
For each patient i, have a vector of features representing their baseline sean x; in R and matrix of additional scans that they received X; in Rdxa, where ni, is the number of their additional scans. Our method learns a new representation for a patient, Y; that is enriched by the additional longitudinal data available for that patient, X₃. We do so by learning projection W in Rdxr from the additionally available data, where r hyperparameter. Figure shows a visual representation of the enrichment we apply to the data. We calculate each individual patient separately so for notation simplicity we will discard the subseript when there is no interaction across patients. The enriched representation is calculated as,
y = W x.
(1)
Baseline
M6
M12
M18
M24
M36
yi
X
Learn a projection W from follow-up scans enrich the common baseline.
yj W Xj
M6
M12
M18
M24
M36
Baseline
Figure 1: Enriching consistent patient data with additional data despite inconsistencies.
Another relevant aspect of data how we consider outliers. Outliers can heavily impact the learned models, especially when we utilize fz-norm distances such as (Wang et al.] [014). Many robust methods, [Wang et al. 2014], have been developed to manage the impact of outlicrs especially when using the squared 22- no-mm distance. Some have chosen to use the fo-norm, fn-nmm, & norm as way to mitigate outliers' disproportionate effect. In our method, we replace the squared order of the distances with parameter p where 0 p 2. Having 1&2 allows us to control and tune the impact of outliers in our model in data-specific way. We refer p as the outlier sensitivity. Possessing fine control over the model's sensitivity outliers can be a key performance advantage in various applications such as medical diagnosis.
parameter that includes
OBJECTIVE
The general objective for our method to learn useful projection from additional longitudinal data and then apply the projection to enrich the baseline data. The projection is learned for each patient independently. The and j subscripts reference the individual seans that make up the additional data for each target patient. The projection is comprised of both global and local patterns learned from the additional data. To capture the global patterns, Principal Component Analysis (PCA) creates reduced representation of the longitudinal data that maximizes the explained variance across the
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data. PCA is maximization of,
T4e JPCA Σ
|
wt.W' II
In addition accounting for global patterns within the data, consider maintaining local pairwise patterns using Locality Preserving Projections (LPP) (He & Niyogi 2004). LPP will preserve the closeness and structure of nearby data points after the data has been projected into lower dimen- sional space. Given a pairwise similarity matrix of the additional data, S, LPP preserves the local
relationships and maximizes the smoothness of the manifold of the data in the embedding space by minimizing (He & Niyogi 2004).
72
f2
JIPP
8ig
W
x;
W x, 12-
==
Our goal is to learn a dimensionally reduced representation of the longitudinal data that preserves both the local and global patterns the data by combining the PCA and LPP objectives. We deter- mine projection by solving.
Joobbined = JLPP - aJPCA:
s.t. W W
(4)
where 0 S a S 1 is the tradeoff hyperparameter between the two projections. Empirically, the opti- mal is difficult and cime-consuming caleulate. To avoid calculating alpha, we present quotient formulation that is equivalent in balancing between the objectives while inherently calculating o,
x;
W' x,
min W
s.t. N..WTW W
(5)
x
Solving the objective JLPP results in projection that benefits from both PCA and LPP, while Jupr preserving their constraints
To support the constraint for this objective we rewrite the problem such that the constraint must hold true. We can verify that if W W I the following equation holds,
bb ww b
|b.11
|
b
(6)
by which we can rewrite the problem as,
D Sij
W
W x'xxl x;
min Σ
S.L.W W =I
X, WWTx,
Additionally, since the squared norms are susceptible to outliers, we convert all norms p-order where 0 p 2 to allow for tuned outlier sensitivity. This results in the problem as.
E E
W X; w "1, X,
min W
S.1. W. =I
Σ X
WWT x, 1I
ALGORITHM TO CONVERT RATIO MINIMIZATION TO DIFFERENCE MINIMIZATION
Before solving the problem in Equation 8 first consider the more general problem,
f(v)
min
where g(v) 2 0 (VUEC)
g(v)
We require that the problem presented in Equation be lower bounded, which is true for Equation The algorithm to solve Equation is presented in Algorithm [ In this section present theorems that prove that the algorithm converges to the global optimal solution, and at quadratic convergence rate. The detailed proof of these theorems is provided in the appendix.
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Algorithm 1: Algorithm to solve Equation 9
Initialize a & C;
while not converge do
Calculate a(n)*
f(v).
Update v by solving the following problem:
arg min f(u) >g(v)
(10)
PEC
Theorem (Wang et al. 2014) Algorithm ! decreases the objective value of problem in each
iteration if the updated Equation 10 satisfies f((v) >g(u> 2
Theorem (Wang al.
2014 Algorithm
convergence rate is quadratic.
Theorem 3 If the updated in Equation 10 of Algorithm i is stationary point of problem 10 the converged solution in Algorithm is a stationary point of problem
Theorem Algorithm decreases the objective value of the problem outlined Equation in each iteration if the updated value of in Equation 10 satisfies f(u) >g(v)= S O.
Theorem Algorithm Ts convergence rate is quadratic.
Theorem indicates that Algorithm I converges very fast to either a global minimum (if convex) or local minimum (otherwise), i.e., the difference between the current objective value and the optimal value is smaller than z where e >> is a constant, at the I-th iteration.
Our formulation of the problem in Equation 8 fits into this general problem since the denominator is non-negative. The resulting formulation becomes,
n2g
min
ΣΣ> s/y a W
x;
X
1
||x
ww x111)),
(II)
1=1
s.4. W
Here, we can remove the Σj= ||x term since is a constant for any given data set for a given any iteration, resulting in the problem being,
min W
I x
WWWx,1l
s.t.' =I (12)
SMOOTHED ITERATIVE REWEIGHTED METHOD
The problem in Equation is a non-smooth objective and is difficult to efficiently solve in general. In this section, we introduce the smoothed iterative reweighted method to solve this optimization problem.
First, let consider the general problem,
f(x) Σ tr (r((() (0) (z)g((=)))
min aEC
(13)
When (x(x) is vector output function, tr (6o) (x)) becomes,
(lo (x)) (=)
(14)
Equation 13 is non-smooth, so turn solve the following smooth problem
min ((x Σ tr tr( (x))(()) + 81)
(15)
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where 8 >> O is a small constant. When 8 0, Equation 15 is reduced to Equation 13 since,
lim tr( (x)) 8)) (>))(=) 6=
(16)
We derive Algorithm 2 to solve the problem in Equation 15 with the detailed derivation provided in the appendix. The convergence of Algorithm is guaranteed with the proof in the appendix.
Algorithm 2: Algorithm solve Equation 15
Initialize E EC:
while not converge do
For each i, calculate D, [ (x) + σ1) =
2. Update by solving the problem min (x) I Etr(sf (z))
Output:
We note that the iterative reweighted method introduced in (Candes al. 2008 Nie et al. 2010) solves the non-smooth E -norm or f2 norm minimization problems. However, the method in (Can- des et al. 2008 Nie et al. 2010) does not explicitly the smoothness constant (i.e., ō in Equation 15).
Without the smoothness term, the algorithm is impacted by the singularity problem due to in- verted matrices that divide Os, which routinely lead inferior learning performance. smoothness term was informally used in (Nie et al. 2010: Wang et al. 2013 Nie et al. 2013) to improve nu- merical stability. However, the authors did not include
their proof of the convergence of the algorithm. As theoretical contribution of this paper, we formally introduce the smoothness term (i.e., 81 in Equation 15) in our algorithm and theoretically prove the algorithm's 's convergence in which the smoothness term leads to more stable solutions. We refer to Algorithm 2 as the Smoothed Iterative Reweighted Method. We can represent our problem following this general problem as,
R2
min W
W x;
W
x
+
A
0g |x3 - WwT (VW*x11B) x;1,),,
s.t. W.WWWW-L II
j=1
3=1
(17)
R2
Siz W
W x 112
where x
(18)
14
xx WW x, 11
j=l
73 2
|
x,
W" x, _ 5 ,)
(19)
1=1
0, 2
(I|x;
WW
(20)
We can simplify this problem's presentation by introducing 8ig 7j8ij and 0; =x3 resulting in,
The
min W
W - W x,
+
0, ||x
www x11 s.t. -. (21)
To solve this problem, we follow the iterative algorithm outlined in Algorithm 3 solution to the two-order problem, we rewrite the objective in matrix form. We use I and θ to rep- resent diagonal matrixes in Ra, x with each value representing 7 and 0, respectively. Therefore,
determine
= TS and θ = @A. We also introduce the Laplacian matrix, L = D S where is the diagonal degree matrix with du -En 5ig. We rewrite the problem as,
min tr (WWTTXXX' W
+ (r - WWT X)O - X'WW"))
(22)
= tr (XX Xêx
W), W..
(23)
We directly solve the problem using the Rayleigh quotient (Hom & Johnson 1985) resulting in W being the r smallest eigenvectors of XLXT XĒXT. Our solution algorithm is outlined in Al Algorithm
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Algorithm 3: Algorithm to solve problem outlined in Equation
Initialize W
while not converge do
Update À using Equation
18
while converge do
Update ? using Equation
Update d using Equation
Update W as the smallest r eigenvectors of XLX" -xêx7
end
end
EXPERIMENTS & RESULTS
To empirically support the value of our method, we conduct several experiments. First, we analyze the convergence of our solution algorithm, ensuring that it in fact converges as theoretically proven. Second, we perform a parameter study and conduct classification experiment using our method at varying values of outlier sensitivity, p. We then compare the performance of the best enriched representation after hyperparameter tuning with the baseline representation at predicting patients' diagnoses using classical classification methods. Finally, investigate the interpretability of our method by checking the coefficients allocates to each feature when enriching the data.
For our experiments, used brain imaging scans from the Alzheimer's 's Discase Neuroimaging Ini- tiative (ADNI) 544 patients. The patients in our experiments all had a baseline scan and between 3 and 5 additional scans taken various time-point of at least 6 month increments: 6 months, 12 months, 18 months, 24 months, and 36 months. Each patient's 's diagnosis belongs to one of the following classes: Alzheimers Disease (AD) 92 patients, Mild Cognitive Impairment (MCI) 205 patients, and Healthy Control (HC) 177 patients. We performed voxel-based morphometry (VBM) on the MRI data by following (Risacher al., 2010) and extracted mean modulated gray matter mea- sures for 90 target regions of interest (ROI). These measures are adjusted for the baseline intracranial volume using regression weights derived from the HC participants at baseline. Our selected value for the dimensionality of the enriched representation was set at r = 3 for all experiments.
5.1 ALGORITHM CONVERGENCE ANALYSIS
We monitor the objective function's value over many iterations for several subjects to monitor its convergence. The value of the objective function over time is shown in Figure 2 Each line in the figure represents a different patient, showing different rates of convergence depending on the individual's 's data. The objective generally converges a fast rate of under 10 iterations.
0.25
0.5
1
p 1.5
0.20
Oljective
1.5
Objective
0.4
OBjecivve
1.0
0.10
0.5
0.05
0
10 Iterations
10
Iterations
Iterations
Iterations
Figure Algorithm convergence for 10 example subjects across different values of p.
5.2 PARAMETER STUDY
A powerful attribute of our method's approach is the ability to tune outlier sensitivity, p. Here we monitor the effect of changing the value of p classification performance. We use both accu- racy and weighted F-1 scores to measure the performance our classification experiments that we conduct 10-fold cross validation test, taking into consideration the standard deviation of

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metrics. We run the experiment with Decision Trees, Gaussian Naive Bayes, Multi-Layer Percep- trons (MLP), Support Vector Machines (SVM), and k-Neerest Neighbors (kNN) models following hyperparameter search. The results for accuracy and weighted F-1 scores are shown in Table and 2 respectively where we present the mean score and 95% confidence interval. Bolded results represent the best results after taking into consideration statistical p-values of pairwise T-test when compared to the baseline results. Methods that do not have bold value mean that the results were not significant enough for to confidently state an improved diagnosis performance. The statistical p-values are provided in tables in the appendix.
The best value for outlier sensitivity p can often vary and in this case the most common superior model utilizes p = 0.25. This shows the value in modifying our distance choices from a simple 2-order distance. The outlier sensitivity tuning allows us to find a better result for our model that customizes the weights of outliers versus other points. We believe that models of different appli- cations should account for outliers differently and therefore there is value in providing this model customization. Our results provide supporting evidence to our case.
View
Decision Tree
Naive Bayes
MLP
SVM
6-NN
Baseline
0.455 + 0.107
0.397 0.108
0.496 = 0.113
0.542 = 0.165
0.484 0.124
p=0.25
0.552 + 0.113
0.588 0.0954
0.523 0.0457
0.518
0.0461
0.549 0.108
p=0.5
0.477 + 0,129
0.462 0.0581
0.514 + 0.0091
0.514 + 0.0091
0,48 + 0.0978
Enriched
p=1
0,442 = 0.0766
0.495 + 0.0324
0.514 = 0.0091
0,514 = 0.0091
0,489 = 0,0947
1=
0.46 1. 0.15
0.504 1 0.0245
0.514 1 0.0091
0.514 1 0.0091
0.491 1 0.0924
p=2
0.527 +: 0.0995
0.498 + 0.0657
0.532 + 0.0954
0.59 + 0.12
0.574 + 0.0933
Table I: Accuracy scores for various values of p across different classification methods.
View
Decision Tree
Naive Baycs
MLP
SVM
k-NN
Baseline
0.453 + 0.102
0.372 = 0.099
0.331 + 0.113
0.48 + 0.211
0.444 + 0.12
p=0,25
0.55 = 0.0615
0.539 + 0.0831
0.389 = 0.129
0.383 - 0.0658
0.532 + 0.111
0-5
0.474 1: 0.128
0.417 0.0687
0.35 1 0.0103
0.381 1: 0.0465
0.437 1 0.118
Enriched
p=1
0.433 + 0.101
0.354 = 0.0458
135=000100
0.35 0.0103
0.45 + 0.0973
p=1.
0.455 0.158
0.353 0.0234
0.384 0.101
0.35 0.0003
0.458 + 0.126
p=2
0.518 0.0882
0.367 = 0.0634
0.439 = 0.221
0.569 + 0.114
0.564 + 0.0894
Table 2: Weighted F-1 scores for various values of p across different classification methods.
5.3 SUCCESSFUL DIAGNOSIS RATE COMPARISON
Following our parameter study, we compare the performance of the classification models the baseline data to the performance of the models using the data longitudinally enriched with our learned projections. We also present the statistical p-value of the T-test for paired samples given that we used the same folds across experiments. The resulting scores presented in Table [ and 2 also compare our method's performance to the baseline representation. Our method significantly outper- forms the baseline representation when used with Decision Trees, Naive Bayes, and k-NN, showing the value added from the available additional data for patients and the enriched representation.
5.4 MODEL INTERPRETABILITY
A key feature in our method's 's approach interpretability in terms of the key biomarkers that dif- ferentiate the various patient scans. Including PCA in the projection allows us to use the learned coefficients of the projection, W, to determine the importance of key biomarkers. Since we have a projection per patient, we can determine the importance of each biomarker for that patient in partic- ular. We conduct a brief summary of all the patients' projections to determine the most important biomarkers across patients. We tallied the 10 most frequently appearing regions of interest from each patient's 's top 10 projection coefficients and normalized the results to determine the relative weights of each coefficient. These results presented the key unsupervised biomarkers that are learned from the data. These biomarkers are key to differentiating global and local patterns in the data across subjects in the ADNI without any supervision regarding their medical condition. We also repeated
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VRM Unsupervised Top 10 p-0.25
60M4
VBM Supervised Best Fl Top p-0.25
6002
C033
0005
Figure 3: Key biomarkers learned from unsu- pervised patients' projections.
Figure 4: Key biomarkers learned from AD classification.
this analysis by using the best performing decision tree model's 's feature weights multiplied by each patient's projection coefficient determine the weights with respect to classifying AD. This second approach provides a supervised approach to relating key biomarkers with AD specifically. Results from both the unsupervised and supervised analyses are shown in Figure 3 and respectively.
We investigated the top biomarkers in our enriched feature representation and determine their rela- tionship to AD. The following biomarkers were found have the highest relevance to the enriched representation when weighted by the best performing decision tree model's feature importances; putamen, thalamus, amygdala, hippocampus, caudate nucleus, and Heschl's gyrus. We find that our results are generally consistent with medical rescarch AD.
The putamen and thalamus are key biomarkers in diffentiating between patients (Bollen et al. 2008
The amygdala is found to be key region affecting AD in (Poulin et al.. 201 Laakso al. 1995).
The left and right hippocampal regions are found to be in the top selected biomarkers by our model with support from (Laakso al. 1995). The caudate nucleus has some supporting evidence from (Jiji et al. 2013) where it was found to exhibit heavy correlation with AD. More recently, (Persson) al. 2017 found more supporting evidence of the relationship.
The extensive support by medical research of key biomarkers our method has determined elicit confi- dence in ourapproach. Our method shows that it can encapsulate key information from various brain regions over time and distill that in a consistent, small dimensional representation. The following biomarkers were found have the highest relevance in patient differentiation by our unsupervised method without weighting for diagnosis prediction; putamen, thalamus, amygdala, parahippocam- pal gyrus, caudate nucleus, olfactory sulcus, fusiform gyrus, insular cortex, middle temporal gyrus, and lingual gyrus. The key biomarkers for AD are still included, however the representation en- compasses more than just the focus on AD. In addition to memory retrieval and processing, the biomarkers represent other brain functions such This allows us to obtain a representation that isn't 't only focused on AD but can represent a patient holistically across time-points regardless of the task to be performed.
vision, olfaction, recognition, and cognition.
6 CONCLUSION
Longitudinal data with inconsistently available sample data can cause difficulty with learning con- sistent data representations. This evident in medical studies such as the Alzheimers Disease Neuroimaging Initiative. Our proposed method calculates a patient-specific projection that allows different patients to have varying additional data and produces consistent-lengtle representation for all patients by enriching the shared baseline with their additional data. Our results show that our en- riched representation surpasses the performance of the baseline representation predicting patient diagnoses.
limitation of our method is that the dimensionality of the enriched representation is limited by the minimum number of additional data patient has, i.e. < minys,, This limitation isn't often impactful due to large amounts of available data for longitudinal problems. In addition to the su- perior performance of predicting diagnoses, our method provides both unsupervised and supervised interpretability. The resulting coefficients of key regions of interest of the brain are consistent with medical research studying the brain and AD's effects. Our method can achieve superior performance by enriching consistent data with additionally available data, thereby more accurately predicting AD and potentially improving or extending many patients' lives.
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A APPENDIX
Before deriving the algorithm to optimize the problem in Equation 15 we need the following proofs and lemmas.
7 we represent the updated in Equation 70 as ē which satisfies f(0) ( Ag(ō)= I f(v)->g(v) Thus f(ō)-}g(ū) S
Proof A.1 In each iteration of Algorithm
which indicates that [ [(1]
g( g(
because g( 2 0.
Proof A.2 The Lagrangian function of the problem in Equation 70 is
a1, denote the converged solution Algorithm 7 8². If o is stationary point of problem 10 we
Li a1, 31) f(v) - >g(u) + a p(v) + 87 (v).
(24)
have
f(v*)
f(0)) g((r) g (v") + + =0
(25)
which can be written as
((v)) - f(u (w))
+ af
g(ve) ('w0))((0') +
(1")
g(0*) o' (ee) O.
(1()))
Let 02 g(v*)
and B2 glery. then we have
B
f00)
+ +
(26)
g(u)
Note that 0n 2 0 and g(0*) 2 so 02 2 Therefore, v- is a stationary point the Lagrangian
function of problem 9 as follows:
f(v)
C2(u, 32) g(t)
+
(27)
which completes the proof.
Proof A.3 We define a function h(A) as.
h(A) min f(o) - (())-) EEC
(28)
According to Algorithml 1 the converged X the root of h(2), that is, h(X*) O. In each iteration of Algorithm 7 we represent the updated v in Equation 10 as e. According to line 4, h(A) f(ō) Ag Thus h' =x)
Using Newton's 's method, the updated solution should be x which is equivalent to step 1 in the next iteration of Algorithm Algorithm i Newton 's method
M())
1(0)-Ag(8) f(0)
W(X
-9(U)
a(
solution to find the root of h()), and its convergence rate quadratic.
Based on the chain rule, we have:
Lemma 1 Suppose g(z) is matrix output function, h(x) is a sealar output function, and is scalar, vector or matrix variable, then
Σ
ah(gta)) [x]
(g)(())
Oh(g(=))
(Bh(y(=))
T 0g(x)
(29)
Ox
dx
Ög(z)
Ör
According to the chain rule in Lemma I
we can derive the following lemmas:
Lemma 2 Suppose g(x) is scalar, vector or matrix output function, and x is a scalar, vector, or matrix variable, then
atr((g" ()))l)())))))))))))))))))))))))) (x)g(z)++615)) (g" (1))(()) 61) 2 T (xx) 0g(x)
(30)
dr
dr
11

---

Under review as a conference paper at ICLR 2020
Lemma Suppose g(z) is scalar, vector matrix output function, is scalar, vector matrix variable, and D is a constant that is synmetrrical if it a matrix, then
0tr(g" (x)g(x) 22D" (x) dr
0g(x) dr
(31)
Now we derive the algorithm to optimize the problem in Equation 15 The Lagrangian function of the problem in Equation 15
C(s, A) = f(x) Eir((o) (<)))(>) + +T)2 x(x,))
(32)
where r(z, A) is a Lagrangian term for the constraint C. By setting the derivative of Equation 32 wrt to zero, we have
0L(,,) A) dz
Str((g] (1)g;(1) + 87)
Ör(z, A) 0
f())
(33)
dx
8x
According to Lemma
2 Equation 33 can be rewritten as
(())) Σ plal (z)g.(x) (())(())
+ ō1) 2 gi (x)
(g)(x) Oz
0r(x, A)
(34)
dx
If we can find solution, that satisfies Equation 34
can find local global optimal solution to the problem in Equation 15 according to the K Karush- Kuhn-Tucke (KKT) conditions (Boyd &
Vandenberghe: 2004). Following (Candes et al. 2008 Nie et al. 20T0). propose iterative
algorithm to find solution to Equation 34 uuiing the observation that D; E (of (z) 2
is a constant. Equation 34 reduced to:
0gi(z) 0r(x, =0 Or Fx
f((x) Σ 2D.9 (x)
(35)
which is equivalent to solving
min ((x Σ tr(gl (c)g(
(36)
Based these observations,
first guess solution to Equation 15
Then calculate D)
using the current value of x and update x by the optimal solution of the problem Equation 36 by
calculating D,- We iteratively perform this procedure until it converges, which is summarized in
Algorithm 2
The convergence of Algorithm 2 is guaranteed by the following theorem. (The proof of Theorem is supplied in the appendix due to space limitations.)
Theorem Algorithm 2 will monotonically decrease the objective of the problem in equation 75 in each iteration until it converges.
During convergence, Equation 34 will hold, thus the KKT conditions of the problem in Equation 15 are satisfied. Therefore, Algorithm 2 will converge to local optimal solution to the problem in
Equation 15 If the problem in Equation 15 is convex, Algorithm 2 will converge global optimal solution.
PROOF OF THEOREM
Before proving the convergence of Algorithm 2
we first introduce several lemmas:
Lemma For any > 0, the following inequality holds when <<PS> 2:
2
2-p
(37)
2
2
12

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Under review as a conference paper at ICLR 2020
Proof. Denoting f()) = po - -2o + 2
p, we have the following derivatives:
(() = p(1 - -a -²²²) and ["(o) p(2 -p -
When o > 0 and 0 p 5 2, (')()) 2 0 and o = 1 is the only point that f'(o) = Note that
f(1) = 0, thus when true.
0 and 0 < S 2, then f(o) 2 0, which indicates Inequality 37 holds
Lemma Ruhe 1970 For any positive definite matrices M, with the same size, suppose the digen-decompositions. M UEUT, M = VAVT, where the eigenvalues E. are increasing order and the eigenvalues in A are in decreasing order. Then the following inequality holds when < p S 2:
(r(MM)& (EA)
(38)
Lemma For any positive definite matrices M, M with the same size, the following inequality holds when 0 p 2:
tr(M) 2 P ) S (1(()) - PIr(MM 42).
(39)
Proof. For any a 0, and O, which indicates
p S according to Lemma we have 2(4) (2)
-
2-P 2
B 2 0 + 2- 2
(40)
2
Suppose the eigen-decomposition M = UEUT M = VAVT, where the eigenvalues in is in increasing order and the eigenvalues in A is in decreasing order. Then according to Inequality 40 we have
P ir (I(AAT tr(=5)+ + 2- 1r((1)22
(41)
2
and according to Lemma we have
( P tEAA _=2 20
(42)
2
2 tr Prr 2- ) - tr(x + 2 2-1 tr 2
(43)
2
Note that (A( tr(23) and tr (r(M)) (A(AA). so we have
5tr 72 tr + 2r)
+ tr(18)) - _tt(MMM 332 < 2r
= tr( Ptr(MM "²") tt((M)) (tr))-(rMMM
which completes the proof.
Lemma For any matrices A, with the same size and > O. the following inequality holds when
0 < p <
(r(( À + 8) (rr(A" A(A" A + 5T) 3
(44)
< (rA A 8) A(A" A+ 81)
Proof. Note that Ä Ā + 57 and AT A + 5T are positive definite matrices since ö > 0. Then
according to Lemma we have
tr (r'' ( 61) A ( 82) 53)
(45)
< tr((AT A+ (1)+)1)5)-11 Str A A 81) -
A+
which indicates that Inequality 44 holds true.
As a result, we can prove Theorem follows now.
In step 2 of Algorithm 2 we denote the updated x: as ã. According step 2, know
[(ā) Σ tr(oT (1)) f(x) Σ tr(g7 (x))
(46)
13

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Under review as a conference paper at ICLR 2020
where the inequality holds when and only when the algorithm converges. For each i, according to Lemma]
we have
+ 01)
(G))) (z) 51)
(47)
I (1)
Etr(o" (z)) (x)g:(x) 51) 5
Note that D. = [o (z)).()) + 57) = so for each i we have tr((g] (T)g. (o) و (T)g(())) tr((g)) (x)g(()) +8) (1))(()))))
(48)
Then we have
Σ Etr(la (T)g 81) Σ tr(a (E)
(49)
s E tr(lol (x) + 87) Σ tr(g (r))()))))
Summing Inequality
46 and Inequality 49 on both sides, we arrive
f(0) Σ tr(lof (z) + +
(50)
((x) + Σ (x)gr())
(c) + 51)5)
Note that Inequality holds holls only when the algorithm converges. Thus the Algorithm 2 will mono- tonically decrease the objective of the problem in Equation T5 in each iteration until the algorithm
converges.
In the convergence, the equality in Equation 34
will hold, thus the KKT condition of problem 15 is
satisfied. Therefore, the Algorithm will converge to local optimum solution to the problem 15 If the problem 15 is convex, the Algorithm 2 will converge to a global optimum solution.
14

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DETAILED EXPERIMENTAL RESULTS
Under review as conference paper at ICLR 2020
The following tables (Tables & include the same results the tables in the original content with the addition of p-values denote the significance of the results when compared to the Baselie method. The results these tables should clarify the reasoning behind the winning bolded results as shown in the original content.
Table 3: Accuracy scores for various values of p across different classification methods
View
Decision Tree
Naive Bayes
MLP
SVM
k--N
Baseline
0.455 0.107,
0.397 + 0.108,
0.496 + 0.113,
0.542 + 0.165,
0.484 + 0.124,
p=255
0.552 0.113, 0.00966
0.588 1 0.0954, 2.92e 06
0.523 + 0.0457, 0.287
0.518 + 0.0461, 0.391
0.549 + 0.108, 0.0203
0=0.5
0.477 0.129, 0.453
0.462 + 0.0581, 0.00795
0.514 1: 0.0091, 0.343
0.514 1 0.0091, 0.344
0.48 0.0978, 0.895
Enriched
p=1
0.442 = 0.0766, 0.553
0.495 1 0.0324, 0.000213
0.514 1: 0.0091, 0.343
0.514 1 0.0091,
0.489 + 0.0947, 0.837
p=1.5
0.46 = 0.15, 0.811
0.504 + 0.0245, 0.000312
0.514 + 0.0091, 0.343
0.514 10.099999.343
0.491 + 0.0924, 0.747
p=2
0.527 0.0995, 0.0292
0.498 = 0.0657, 0.000198
0.532 + 0.0954, 0.144
0.59 + 0.12,
0.574 0.0933, 0.00523
Table 4: Weighted F-1 scores for various values of p across different classification methods
ST
View
Decision Tree
Naive Bayes
ML.P
SVM
k-NN
Baseline
0.453 0.102,
0.372 + 0.099,
0.331 0.113,
0.48 0.211,
0.444 + 0.12,
0=225
0.55 + 0.0615, 0.00057
0.539 + 0.0831, 8.04e 07
0.389 + 0.129, 0.147
0.383 + 0.0658, 0.0341
0.532 + 0.011,000562
0-5
0.474
0.128, 0.439
0.417 + 0.0687, 0.0306
0.35 + 0.35 0.0103, 0.343
0.381 + 0.0465, 0.0252
0.437 + 0.118, 0.844
Enriched
p=1
0.433
0.101, 0.4
0.354 + 0.0458, 0.377
0.35 1 0.0103, 0.343
0.35 1: 0.0103, 0.00508
0.45 1 0.0973, 0.798
p=1.5
0.455
0.158, 0.925
0.353 1 0.0234, 0.33
0.384 1 0.101, 0.0318
0.35 0.0103, 0.0103,0.005000 0.00508
0.458 1 0.126, 0.611
0.518 p=2
0.0882, 0.034
0.367 +1 0.0634, 0.734
0.439 0.221,0.0393
0.569 + 0.114,0.0123
0.564 1 0.0894, 0.00129

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Under review as a conference paper at ICLR 2020
PARAMETERIZED BRAIN PLOTS
In this section we include additional plots from the feature weighting described in Section These plots for various values of p and cover both the unsupervised and supervised model weighting to differentiate between all subjects and subjects with AD respectively, First, we include all the plots from the unsupervised feature extraction that denote the differences across all subjects. Second, we include the plots after weighting the biomarkers by the learned weights from the decision tree models with the best F-1 score at diagnosing patients with AD.
Unsupervised plots
VBM Unsupervised Top 10 p=0.25
0.044
L
R
R
0.033
0.022
0.011
Figure 5: Unsupervised key biomarkers with p = 0.25.
VBM Unsupervised Top 10 0-0.5
0.045
R
0.034
0.023
0.011
Figure 6: Unsupervised key biomarkers with p = 0.5.
16

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Under review as a conference paper at ICLR 2020
VBM Unsupervised Top 10 p=1
0.043
L
R
R
0.032
0.021
0.011
Figure Unsupervised key biomarkers with p =
VBM Unsupervised Top 10 p=1.5
0.037
R
0.028
0.018
0.0092
Figure 8: Unsupervised key biomarkers with p 1.5.
VBM Unsupervised Top 10 p=2
0.04
L
R
0.03
0.02
0.01
Figure 9: Unsupervised key biomarkers with p 2.
17

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Under review as a conference paper at ICLR 2020
Supervised plots
VBM Supervised Best F1 Top 10 p=0.25
0.023
L
R
R
0.017
0.011
0.0057
0
Figure 10: Supervised key biomarkers for AD diagnosis with 0.25.
VBM Supervised Best F1 Top 10 p=0.5
0.024
L
R
0.018
0.012
0.0061
0
Figure 11: Supervised key biomarkers for AD diagnosis with = 0.5.
VBM Supervised Best F1 Top 10 p=1
R
0.019
0.013
0.0064
Figure 12: Supervised key biomarkers for AD diagnosis with p
18

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Under review as a conference paper at ICLR 2020
VBM Supervised Best F1 Top 10 1.5
0.024
R
0.018
0.012
0.006
Figure 13: Supervised key biomarkers for AD diagnosis with p 1.5.
VBM Supervised Best F1 Top 10 p=2
0.031
R
0.023
0.015
0.0077
Figure 14: Supervised key biomarkers for AD diagnosis with p 2.
19


Reviews:
[{"review_id": "SkgHvwtCFr", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This manuscript develops a metric-learning with non-Euclidean error terms for robustness and applies in to data reduction to learn diagnostic models of Alzheimer's disease from brain images. The manuscript discusses a metric-learning formulation as a quotient for reconstruction-error terms, how to optimize the quotient based on results from Wang et 2014, an iterated reweighted approach to circumvent the non-smooth part of the l1 loss in zero, and experiments on brain images of Alzeimer's disease.\n\nMy two main issues with the manuscript is that the theoretical part is written very imprecisely and that the experiments are not convincing due to the lack of good baselines and of statistical power.\n\nWith regards to the theoretical contributions, a fraction of the results in the present manuscript are trivial consequences or Wang2014, and yet it comes with errors in the statements. For instance, in equation (9), the present manuscript writes greater or equal, while I believe that it should be strictly greater, as in Wang. Theorem 1 and 4 seem almost the same thing, though with a contradiction between the two. Other statements are inaccurate: the authors claim some results on reaching global optima, while I believe that they can only claim that they reach stationary points. Theorem 2 and 5 seem to be the same thing.\n\nConcerning the iterated reweighted approach, I believe that this is non smooth only for g(x)=0, which is not covered by the theorems of Wang 2014. Is this algorithm needed? Note that Wang apply their algorithm with an l1 norm, ie non-smooth in zero, and do not report problems with out. The manuscript mentions that \"to inverted matrices that divide 0s, which routinely lead to inferior learning performance.\". I am not exactly sure what that means and I would need to understand better the problem. Also, the theoretical contribution that with the added the delta the algortihm converges, seems quite trivial: it seems to me that it is the eta trick.\n\nMinor comments: in algorithm 3, it would be useful to write the full expression of the equations, rather than just reference the numbers. Also, the computational cost of the eigenvectors at each iteration seems quite prohibitive.\n\nHow was the value r=3 selected?\n\nFigure 2 seem to choose that approaches have not converged: they final value is larger than intermediate values?\n\nHow were the p-values between cross-validation assessment of estimators computed? If it was done using standard paired t-test, this is incorrect are the folds are not independent.\n\nWith regards to the experiments, I worry that the model is not compared against simple baselines, such as a PCA.", "extra_scores": {"experience_assessment": "I have published in this field for several years.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "SJlOnavpqB", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposed an unsupervised method to make better usage of the inconsistent longitudinal data by minimizing the ratio of Principal Components Analysis and Locality Preserving Projections.\n\nComments:\n1. It is out of the area of my expertise, but the innovation of combining the PCA and LPP to produce the dimensionally reduced representation of the longitudinal data seems not significant enough, especially when the author did not mention why they chose these two methods out of all the available projection/hashing approaches.\n\n2. The paper is poorly written. Theorem 1 and Theorem 4 state the opposite conditions for v. Theorem  2 and Theorem 5 are exactly the same. If the same theorem has already been proved by previous work [Wang et al., 2014], it is not necessary to prove it again in this paper. The most important theorem which is claimed as the theoretical contribution of this paper (i.e. Theorem 6) did not even appear in the main contents but only in the appendix. The proof of theorems in appendix, Proof A.1, A.2, A.3 did not state which theorem they are corresponding to and are organized in different order as Theorem 3-5, make it hard to read.\n\n3. There are many grammar errors and sentences which are not consistent with English writing conventions. E.g. “This problem is heavily studied with (Wang et al., 2012; Brand et al., 2018; Lu et al., 2018) as example approaches which, despite their effectiveness, present an added complexity to the problem.” is better written as “This problem is excessively studied by (Wang et al., 2012; Brand et al., 2018; Lu et al., 2018) as example approaches which, despite their effectiveness, present an additional complexity to the problem”; “... to provide value in predicting patients’ diagnosis.” is better written as “... to provide valuable information in ...”; “... , we focus AD diagnosis from MRI scans ...” should be “... , we focus on AD diagnosis from MRI scans ...”; “consider outliers” is better as “handle outliers”; etc. \n\n4. In the sentence below equation (5), “J_LPP/J_LPP” should be “J_LPP/J_PCA”. \n", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper."}, "extra_text": {}}, {"review_id": "HygsE0WXqB", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The motivation of this work is to address the issue of inconsistent and missing data in longitudinal clinical studies. \n\nThe main goal of this work is to learn a dimensionality reduction representation by combining locality preserving projection (LPP) and and the global pattern by principal component analysis (PCA). The balance between these two components is modulated by a hyperparameter alfa. The authors proposed a method to overcome the issue of a direct computation of alfa. The main contribution of this work is to prove that a quotient formulation between LPP and PCA is equivalent to a weighted difference of these two terms.\n\nIt is not straightforward to understand whether the proposed method is effective in reducing the time consuming effort required by the computation of alfa hyperparameter, as stated in the premises as motivation of this work.\n\nThe presentation of theorems is misleading. Theorems 1 and 4 seem to claim opposite statements, while theorems 2 and 5 have the same claim.\n\nMinor comments.\nAfter equation 5 there is a typos in the ratio LPP/LPP.\n", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I did not assess the derivations or theory."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes to overcome the issue of inconsistent availability of longitundinal data via the combination of leveraging principal components analysis and locality preserving projections. All three reviewers express significant reservations regarding the technical writing in the paper. As it stands, this paper is not ready for publication. 

Decision: Reject