Ceiling effect is a measurement limitation that occurs when the highest possible score on a measurement instrument is reached, thereby decreasing the likelihood that the testing instrument has accurately measured the intended domain (Salkind, 2010). The ceiling effect has long been discussed across a wide range of scientific fields such as sociology (DeMaris, 2004), educational science (Kaplan, 1992; Benjamin, 2005), biomedical research (Austin and Brunner, 2003; Cox, 1984), and health science (Austin et al., 2000; Catherine et al., 2004; Voutilainen et al., 2016; Rodrigues et al., 2013), because it is a crucial information deficit known to inhibit effective statistical analysis (Austin and Brunner, 2003).
The ceiling effect is also conceivable in the context of machine learning, e.g., in recommendation systems with a five-star rating. After rating an item with a five-star, a user may find another item much better later. In this case, the true rating for the latter item should be above five, but the recorded value is still a five-star. As a matter of fact, we can observe right-truncated shapes indicating ceiling effects in the histograms of well-known benchmark data sets for recommendation systems, as shown in Figure2.
Restoring data from ceiling effects can lead to multiple benefits in many fields. The recovered data may provide us with further findings in the case of scientific research that suffers from ceiling effects in its measurements. The recommendation system may be able to find latent superiority or inferiority between items with the highest ranking and predict unobserved entries better.
In this paper, we investigate methods for restoring a matrix data from ceiling effects, which is a potential novel remedy for ceiling effects. In particular, we consider the recovery of a clipped matrix, i.e., values are clipped at a predefined threshold in advance to observation, because ceiling effects are often modeled as a clipping phenomenon (Austin and Brunner, 2003).
1.1 Our problem: clipped matrix completion (CMC)
We consider the recovery of a low-rank matrix whose observations are clipped at a predefined threshold (Figure 2). We call this problem clipped matrix completion (CMC). Let us first introduce its background, low-rank matrix completion.
Low-rank matrix completion (MC) aims to recover a low-rank matrix from various information deficits, e.g., missing, noise, or discretization (Candès and Recht, 2009; Candès and Plan, 2010; Recht, 2011; Chen et al., 2015; Davenport et al., 2014; Lan et al., 2014; Bhaskar, 2016)
. The principle that enables low-rank MC is the dependency among entries of a low-rank matrix; each element can be expressed as an inner product of the latent feature vectors of the corresponding row and the column. With the principle of low-rank MC, we may be able to recover the entries of a matrix from a ceiling effect.
Clipped matrix completion (CMC)
The clipped matrix completion (CMC) problem is illustrated in Figure 2. It is a problem to recover a low-rank matrix from random observations of its entries.
Formally, the goal of CMC in this paper can be stated as follows. Let be the ground-truth low-rank matrix where , and be the clipping threshold. Let be the clipping operator that operates on matrices element-wise. We observe a subset of entries of . The set of observed indices is denoted by . The goal of CMC is to accurately recover from , and .
The limitations of MC.
There are two limitations regarding the application of existing MC methods to the CMC problem.
The applicability of the principle of low-rank MC to clipped matrices is non-trivial, because clipping occurs depending on the underlying values whereas the existing theoretical guarantees of MC methods presume the information deficit (e.g., missing or noise) to be independent of the values (Bhojanapalli and Jain, 2014; Chen et al., 2015; Király et al., 2015; Liu et al., 2017).
The goal of this paper is to overcome these limitations and to propose low-rank completion methods suited for CMC.
1.2 Our contribution and approach
From the perspective of MC research, our contribution is three-fold.
1) We provide a theoretical analysis to establish the validity of the low-rank principle in CMC (Section 2).
To do so, we provide an exact recovery guarantee: a sufficient condition for a trace norm minimization algorithm to perfectly recover the ground truth matrix with high probability. As the first step, our analysis is based on the notion of incoherence (Candès and Recht, 2009; Recht, 2011; Chen et al., 2015).
2) We propose practical algorithms for CMC (Section 3) and provide an analysis of the recovery error (Section 4).
We propose practical CMC methods which are extensions of the Frobenius norm minimization that is well used for MC (Toh and Yun, 2010)
. The simple idea of extension is to replace the squared loss function with the squared hinge loss to reduce the penalty of over-estimation on clipped entries. We also propose a regularizer consisting of two trace norm terms, which is motivated by a theoretical analysis of a recovery error bound.
3) We conducted experiments using synthetic and real-world data to demonstrate the validity of the proposed methods (Section 6).
Using synthetic data with known ground truth, we confirmed that the proposed CMC methods can actually recover randomly-generated matrices from clipping. We also investigated the improved robustness of CMC methods against the existence of clipped training entries in comparison with MC methods. Using real-world data, we conducted two experiments to validate the effectiveness of the CMC methods.
1.3 Additional notation
For commonly-used notation, please see Table 4 in Appendix. The symbols , and are used throughout the paper. Let be the rank of . The set of observed clipped indices is denoted by . Given a set of indices , we define its projection operator by , where denotes the indicator function giving if the condition is true and otherwise.
2 Feasibility of the CMC problem
As noted earlier, it is not trivial if the principle of low-rank MC is guaranteed to recover clipped matrices. In this section, we establish that the principle of low-rank completion is still valid for some matrices by providing a sufficient condition under which an exact recovery by trace norm minimization is achieved with high probability.
We consider a trace-norm minimization for CMC
2.1 Definitions and intuition of characteristic quantities
Here, we define the quantities required for stating the theorem. The quantities reflect the difficulty of recovering , therefore the sufficient condition stated in the theorem will be that these quantities are small enough. Let us begin with the definition of coherence that captures how the information of a matrix is distributed around its entries (Candès and Recht, 2009; Recht, 2011; Chen et al., 2015).
Def. 1 (Coherence and joint coherence (Chen et al., 2015)).
Let have a skinny singular value decomposition
have a skinny singular value decomposition. We define
where () is the -th (resp. -th) row of (resp. ). Now the coherence of is defined by
In addition, we define the following joint coherence
The feasibility of CMC depends upon the amount of information that the clipping can hide. To characterize the amount of information obtained from observations of , we define a subspace that is used in the existing recovery guarantees for MC (Candès and Recht, 2009).
Def. 2 (The information subspace of (Candès and Recht, 2009)).
Let be a skinny singular value decomposition ( and ). We define
where are the -th column of and , respectively. Let and denote the projections onto and , respectively.
Using , we define the quantities to capture the amount of information loss due to clipping, in terms of different matrix norms representing different types of dependencies. To express the factor of clipping, we define a transformation on that describes the amount of information left after observation. Therefore, if these quantities are small, it is implied that enough information for recovering is preserved after clipping.
Def. 3 (The information loss measured in various norms).
where the operator is defined by
In addition, we define the following quantity that captures how much information owes to the clipped entries of . If this quantity is small, it is implied that enough information of is left in non-clipped entries.
Def. 4 (The importance of clipped entries for ).
We follow Chen et al. (2015) to assume the following observation scheme. As a result, it amounts to assuming that is a result of random sampling where each entry is observed with probability independently.
Assumption 1 (Assumption on the observation scheme).
Let . Let and . For each , let be a random set of matrix indices that were sampled according to independently. Then, was generated by .
2.2 The theorem
We are now ready to state the theorem.
Theorem 1 (Exact recovery guarantee for CMC).
The proof and the precise expressions of and are available in Appendix C. The characteristic quantities (Def. 3 and Def. 4) do not appear in either the order of or that of , but they appear as coefficients and deterministic conditions that enable the theorem to hold. The existence of a deterministic condition is in accordance to the intuition that an all-clipped matrix can never be completed no matter how many entries are observed.
3 Practical algorithms
In this section, we introduce practical algorithms for CMC (clipped matrix completion). The trace norm minimization (Eq. (1)) is known to require impractical running time as the size of the problem increases from small to moderate or large (Cai et al., 2010).
A popular method for matrix completion is to minimize the Frobenius norm between the predicted matrix and the observed matrix, under some regularization (Toh and Yun, 2010). We develop our CMC methods from this approach.
Throughout this section, generally denotes an optimization variable, which may be further parametrized by (where for some ). Regularization terms are denoted by , and regularization coefficients by .
Frobenius norm minimization for MC.
In the MC methods based on the Frobenius norm minimization (Toh and Yun, 2010), we define
and obtain the estimator by
The problem in using this method for CMC is that it is not robust to clipped entries as the loss function is designed under the belief that the true values are close to the observed values. We extend this method for CMC with a simple idea.
The general idea of extension.
The general idea of extension is not to penalize the estimator on clipped entries when the predicted value exceeds the observed value. Therefore, we modify the loss function to
where is the squared hinge loss, which does not penalize over-estimation. Then we obtain the estimator by
Double trace norm regularization.
We first propose to use . For this method, we conducted a theoretical analysis of the recovery error, which is provided in Section 4. For the optimization, we employ an iterative method based on subgradient descent (Avron et al., 2012). Even though the second term, , is a composition of a nonlinear mapping and a non-smooth convex function, we can take advantage of its simple structure to approximate it with a convex function of whose subgradient can be calculated for each iteration. We refer to this algorithm as DTr-CMC (Double Trace-norm regularized CMC).
Trace norm regularization.
Trace norm regularization is a method to relax the trace norm minimization (Eq. (1)) by replacing the exact constraints by the quadratic penalties (Eq. (2) for MC and Eq. (4) for CMC). For the optimization, we can employ an accelerated proximal gradient (APG) algorithm proposed by Toh and Yun (2010), by taking advantage of the differentiability of the squared hinge loss. We refer to this algorithm as Tr-CMC (Trace-norm-regularized CMC), in contrast to Tr-MC (its MC counterpart; Toh and Yun, 2010).
Frobenius norm regularization.
This method first parametrizes as and use for regularization. A commonly used method for the optimization in the case of MC is the alternating least squares (ALS) method (Jain et al., 2013). Here, we employ an approximate optimization scheme motivated by ALS for our experiments. We refer to this algorithm as Fro-CMC (Frobenius-norm-regularized CMC), in contrast to Fro-MC (its MC counterpart; Jain et al., 2013).
|DTr-CMC||Sq. hinge||Tr + Tr||SUGD|
4 Theoretical analysis for DTr-CMC
In this section, we provide a theoretical guarantee for DTr-CMC. Let be the hypothesis space
for some and . Here, we analyze the estimator
The minimization objective of Eq. (6) is not convex. However, it is upper bound by the convex loss function (Eq. (4)) (the proof is provided in Appendix A.1). Therefore, DTr-CMC can be seen as a convex relaxation of Eq. (6) with the constraints turned into regularization terms. To state our theorem, we define the unnormalized coherence of a matrix.
Def. 5 (Unnormalized coherence).
Here, we consider unnormalized coherence defined by
using and from Def. 1. Here we use an unnormalized definition for ease of notation.
Now we are ready to state our theorem.
Theorem 2 (Theoretical guarantee for DTr-CMC).
Suppose that , and that is generated by independent observation of entries with probability . Let , and be a solution to the optimization problem Eq. (6). Then there exist universal constants and , for which with probability at least we have
5 Related work
In this section, we describe related work from the literature on matrix completion and that on ceiling effects. Table 2 provides a brief comparison of the related work on matrix completion.
5.1 Matrix completion methods.
Our feasibility analysis in Section 2 followed the approach of Recht (2011) while basing some details of the proof on Chen et al. (2015). There is further research to weaken the assumption of the uniformly random observation (Chen et al., 2015; Bhojanapalli and Jain, 2014). For simplicity, we omit such extensions. Nevertheless, we believe it is relatively easy to incorporate such additional factors into our theoretical analysis.
Our problem setting of clipping can be related to quantized matrix completion (Lan et al., 2014; Bhaskar, 2016). Lan et al. (2014) and Bhaskar (2016) formulated a probabilistic model which assigns discrete values according to a distribution conditional on the underlying values of a matrix. Bhaskar (2016)
provided an error bound for restoring the underlying values, assuming that the quantization model is fully known. The model of Q-MC can provide a different formulation for ceiling effects than ours by assuming the existence of latent random variables. However, Q-MC methods require the data to be fully discrete(Lan et al., 2014; Bhaskar, 2016). Therefore, neither the method nor the theory can be applied to real-valued observations. On the other hand, our methods and theories allow observations to be real-valued. We believe that the ceiling effect is worth studying independently from quantization, since the data analyzed under ceiling effects are not necessarily discrete.
The use of the Frobenius norm for MC has been studied for MC from noisy data (Candès and Plan, 2010; Toh and Yun, 2010). Our algorithms are based on this line of research, while extending them for CMC.
Methodologically, (Mareček et al., 2017) is closely related to our Fro-CMC. Mareček et al. (2017) considered completion of missing entries under “interval uncertainty” which yield interval constraints indicating the ranges in which the true values should reside. They employed the squared hinge loss for enforcing the interval constraints in their formulation, hence coinciding with our formulation of Fro-CMC. There are a few key differences between their work and ours. First, our motivations are quite different. We are analyzing a different problem than theirs. They considered completion of missing entries with robustness, whereas we considered recovery of clipped entries. Secondly, they did not provide any theoretical analysis of the problem. We provided an analysis by specifically looking at the problem of clipping. Lastly, as a minor difference, we employed an ALS-like algorithm whereas they used a coordinate descent method (Mareček et al., 2017; Marecek et al., 2018), as we found the ALS-like method to work well for moderate sized matrices.
5.2 Related work on ceiling effects.
From the perspective of dealing with ceiling effects, the present paper adds a novel potentially effective method for the analysis of data affected by a ceiling effect.
Ceiling effect is also referred to as censoring (Greene, 2012) or limited response variables
limited response variables(DeMaris, 2004). In this paper, we used “ceiling effect” to represent these phenomena. In econometrics, the de facto standard of dealing with ceiling effects is to use the Tobit model (Greene, 2012). In Tobit models, a censored likelihood function is modeled, and is maximized with respect to the parameters of interest. Although this method is justified by the theory of M-estimation (Schnedler, 2005; Greene, 2012), it does not automatically guarantee a use for matrix completion. In addition, Tobit models require strong distributional assumptions. This is problematic especially if the distribution cannot be safely assumed.
|Type of deficit||Related work|
|Missing||Candès and Recht (2009) etc.|
|Noise||Candès and Plan (2010) etc.|
|Quantization||Bhaskar (2016) etc.|
6 Experimental results
In this section, we show the results for numerical experiments to compare the proposed CMC methods to the MC methods to demonstrate the effectiveness of our approach.
6.1 Experiment with synthetic data
We conducted an experiment to recover randomly generated data from clipping. The primary purpose of the experiment was to confirm that the principle of low-rank completion is still effective for the recovery of a clipped matrix, as indicated by Theorem 1. Additionally, within the same experiment, we investigated how sensitive the MC methods are to the disturbance of the existence of clipped training entries by looking at the growth of the recovery error on non-clipped test entries in relation to increased rates of clipping.
Data generation process.
We randomly generated non-negative integer matrices in that are exactly rank- and of the same magnitude parameter (see Appendix B). The generated data was randomly split into three with ratio , then the first part was clipped at the threshold , to generate the training (), the validation (), and the testing () matrix, respectively. We iterated over , and was fixed at .
We used the relative root mean square error (rel-RMSE) as the evaluation metric, and we considered a result as a good recovery when the error is of order(Toh and Yun, 2010)
. We separately reported the rel-RMSE on two sets of entries: the whole indices, and the non-clipped test entries. For tuning of hyperparameters, we used the rel-RMSE on validation indices:. We reported the mean of five independent runs. The clipping rate was calculated by the ratio of entries of above .
We evaluated the proposed CMC methods (DTr-CMC, Tr-CMC, and Fro-CMC) and their MC counterparts (Tr-MC and Fro-MC). We also applied MC methods after ignoring all clipped training entries (Tr-MCi and Fro-MCi, with “i” standing for “ignore”). While this treatment wastes some training data, it may improve the robustness of MC methods against the existence of clipped training entries.
Result 1: the validity of low-rank completion.
In Figure (3fig:experiment:synthetic:1), we show the rel-RMSE for different clipping rates. The proposed methods successfully recover the true matrices with very low error of order even when half the observed training entries are clipped. One of them (Fro-CMC) was able to recover the matrix after the clipping rate was above . This may be explained in part by the fact that the synthetic data were exactly low rank, and that the correct rank was in the search space of the bilinear model of the Frobenius norm based methods.
Result 2: the robustness against the existence of clipped training entries.
As seen in Figure (3fig:experiment:synthetic:2), the test error of recovery by MC methods on non-clipped entries increased with the rate of clipping. This indicates the disturbance effect for MC methods due to the existence of the clipped training entries. The MC methods with ignoring the clipped training entries (Tr-MCi and Fro-MCi) were also prone to increasing test error on non-clipped entries in the region of high clipping rates, most likely due to wasting too much information. On the other hand, the proposed methods show an improved profile of growth, indicating an improved robustness.
6.2 Experiments with real-world data
We conducted two experiments using real-world data. The difficulty of evaluating CMC with real-world data is that there are no known ground truths, i.e., the true values unaffected by the ceiling effect. Therefore, instead of evaluating the accuracy of recovery for the ground-truth (which is unavailable), we evaluated the performance of distinguishing entries with the ceiling effect and those without. Therefore, to measure the performance of CMC methods, we considered two binary classification tasks in which we predict whether held-out test entries are of high ratings. The tasks are reasonable, because the purpose of a recommendation system is usually to predict which entries have high scores.
Preparation of data sets.
We used the following benchmark data sets of recommendation systems.
with a stride of(approximately 99.0% missing). For ease of comparison with other data, we doubled the values of the data so that the ratings are integers from to .
Movielens (100K)222http://grouplens.org/datasets/movielens/100k/ consists of ratings from 943 users to 1,682 movies on an integer scale from to (approximately 94.8% missing).
Task 1: using artificially clipped training data
In the first task, we artificially clipped the training data and predicted whether the test entries were above the threshold. We artificially clipped the training entries at the threshold . We used for FilmTrust, and for Movielens. We then predicted whether the ratings of the test entries were above the threshold. For prediction, we set the prediction threshold at , and predicted positively for entries above and negatively otherwise.
Task 2: using raw data
In the second task, we used the raw training data and predicted whether the test entries were the maximum value. For running CMC methods, we treated the maximum value of the rating as , i.e., for FilmTrust, and for Movielens. We then predicted whether the ratings of the test entries were the maximum value. For prediction, we set the prediction threshold at , and predicted positively for entries above and negatively otherwise.
Protocols and evaluation metrics.
In both experiments, we first split the observed indices of the raw data with ratio , which were used as training, validation, and test indices. Then for the first task, we artificially clipped the training data at . If a user or an item had no training entries, we removed them from all matrices.
We measured the performance by the f score. Hyperparameters were selected according to the f
score on the validation entries. We reported the mean and the standard error after five independent runs.
We compared the proposed CMC methods with the corresponding MC methods. The uninformative baseline in these experiments (indicated as “baseline”) is to predict positive for all entries, for which the recall is and the precision is the ratio of the positive class.
Result for task 1
The results are compiled in Table 3. By comparing the results between the CMC methods and their corresponding MC methods, we conclude that CMC methods have improved the ability to recover clipped values in real-world data as well. Considering that the completion methods are not designed to optimize the accuracy on the high values, we regard it is acceptable that some MC methods scored below the baseline.
Result for task 2
The results are compiled in Table 3. The CMC methods show improved performance for predicting entries of the maximum value of rating than MC methods. Considering that the only difference between the improved CMC methods and the corresponding MC methods is the use of squared hinge loss function, we regard this is an indication of an improved robustness against the existence of the clipped training entries.
One interesting fact is that we obtain the performance improvement by only changing the loss function to be robust to ceiling effects and without adding extra complexity to the model (such as introducing an ordinal regression model).
|Data||Methods||Task 1 f||Task 2 f|
|Film||DTr-CMC||0.47 (0.01)||0.46 (0.01)|
|Trust||Fro-CMC||0.35 (0.01)||0.40 (0.01)|
|Fro-MC||0.27 (0.01)||0.35 (0.01)|
|Tr-CMC||0.36 (0.00)||0.39 (0.00)|
|Tr-MC||0.22 (0.00)||0.35 (0.01)|
|(baseline)||0.41 (0.00)||0.41 (0.00)|
|Movielens||DTr-CMC||0.39 (0.00)||0.38 (0.00)|
|(100K)||Fro-CMC||0.41 (0.00)||0.41 (0.01)|
|Fro-MC||0.21 (0.01)||0.38 (0.01)|
|Tr-CMC||0.40 (0.00)||0.40 (0.00)|
|Tr-MC||0.12 (0.00)||0.38 (0.00)|
|(baseline)||0.35 (0.00)||0.35 (0.00)|
In this paper, we showed the first result of exact recovery guarantee in the novel problem of clipped matrix completion. We proposed practical algorithms as well as a theoretically-motivated regularization term. We showed the effectiveness of the proposed method, and that the CMC methods obtained by modifying MC methods are more robust to clipped data, through numerical experiments. An important future work is to consider a specialization of our theoretical analysis for the case of discrete data to analyze the ability of Q-MC methods for recovering discrete data from ceiling effects.
TT would like to thank Ikko Yamane, Han Bao, and Liyuan Xu, for helpful discussions. MS was supported by the International Research Center for Neurointelligence (WPI-IRCN) at The University of Tokyo Institutes for Advanced Study.
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Appendix A Details of the algorithms
Here we describe instances of CMC (clipped matrix completion) algorithms in detail. A more general description can be found in Section 3 of the main article.
a.1 Details of DTr-CMC
The optimization method for DTr-CMC based on subgradient descent [Avron et al., 2012] is described in Algorithm 1. In the algorithm, we let denote a skinny singular value decomposition subroutine, and the all-one matrix.
Derivation of the algorithm.
Let denote the Hadamard product. The second regularization term of DTr-CMC can be rewritten as
where is defined by . This is a composition of a non-smooth convex function and a nonlinear operator , hence it is not trivial to find a method to minimize this function. Here, in order to minimize the objective function, we employ an iterative scheme to approximate this function with one that has a known subgradient. For each iteration
, we find a subgradient of the following heuristic objective at:
and update the parameter in the descending direction. This function is a combination of the trace norm and a linear transformation. A subgradient can be calculated by first performing a singular value decomposition, and then calculating .
While we expect the regularization term to encourage the recovery of the values above the threshold, the task is difficult as it requires extrapolating the values to outside the range of any observed entries. To compensate for this difficulty, we initialize the parameter matrix with values strictly above the threshold. This allows the algorithm to start from a matrix whose values are above the threshold and simplify the hypothesis. Therefore, in the experiment, we initialized all elements of with (here, we used reflecting the spacing between choices on the rating scale of the benchmark data of recommendation systems. This value can be arbitrarily configured).
Range of hyperparameters.
In the experiments, we used . The regularization coefficients and were grid-searched from .
Relation to the theoretical analysis in Section 4.
By a simple calculation,
Therefore, is an upper bound of the objective function of Eq. (6). ∎
a.2 Details of Tr-CMC
For trace-norm regularized clipped matrix completion, we used the accelerated proximal gradient singular value thresholding algorithm (APG) introduced in [Toh and Yun, 2010]. APG is an iterative algorithm in which the gradient of the loss function is used. Thanks to the differentiability of the squared hinge loss function, we are able to use APG to minimize the CMC objective (Eq. (4)) with . We obtained an implementation of APG for matrix completion from http://www.math.nus.edu.sg/m̃attohkc/NNLS.html, and modified the code for Tr-CMC.
a.3 Details of Fro-CMC
This method first parametrizes as , where , and minimizes Eq. (5) with . Here we use to denote the (transposed) row vectors of and .
The minimization objective is not jointly convex in . Nevertheless, it is separately convex when one of and is fixed. The idea of alternating least squares (ALS) is to fix when minimizing Eq. (3) with respect to , and vice versa. In its original form, each update is analytic and takes the form
The squared hinge loss is differentiable, and its derivative is . Thanks to its differentiability, the loss function Eq. (4) is also differentiable. However, in the case of CMC, a closed-form minimizer is not obtained as in the derivation of the original ALS, due to the existence of indicator function in its derivative. As an alternative, we derive a method to alternately update the parameters by minimizing an approximate objective at each iteration. Denoting , we use the following heuristic update rules to approximately obtain the minimizers.
is the identity matrix. We iterate between Eq. (8a) and Eq. (8b) as in Algorithm 2. For the same reason as DTr-CMC, we let the algorithm start from a matrix whose values are all . In the algorithm, we let and be the all-one matrices.
In the experiments, hyperparameters were grid-searched from , and .
|Ground truth matrix (rank )|
|Clipped ground truth matrix|
|Observed clipped entries|
|Subspace of matrices spanned by products of singular vectors of|
|Skinny singular value decomposition of|
|The row-wise (column-wise) coherence parameters|
|Joint coherence of|
|The set of feasible perturbations|
|The importance of for|
|The Lipschitz property of w.r.t. the Frobenius norm|
|The Lipschitz property of w.r.t. the infinity norm|
|The Lipschitz property of w.r.t. the operator norm|
|The number of partitions that generated (introduced for the theoretical analysis)|
|The loss function for CMC using only the squared loss|
|The loss function for CMC using both the squared loss and the squared hinge loss|
|The set of real numbers|
|The set of natural numbers|
|Landau’s asymptotic notation for|
|Element of the matrix|
|Projection to ();|
|the linear operator to set matrix elements outside to zero|
|The Euclidean norm of vectors|
|The trace norm|
|The operator norm|
|The Frobenius norm|
|The entry-wise infinity norm|
|The subspace of spanned by|
|The range of a mapping|
Appendix B The generation process of the synthetic data
In order to obtain rank- matrices with different rates of clipping, synthetic data were generated by the following procedure.
For a fixed , we first generated a matrix
whose entries were independent samples from a uniform distribution over.
We used a non-negative matrix factorization algorithm [Lee and Seung, 2001] to approximate with a matrix of rank at most .
We repeated the generation of until the rank of was exactly . Note that with this procedure, may become larger than .
We randomly split into with ratio , which were used for training, validation and testing, respectively.
We clipped at the clipping threshold to generate and removed entries randomly with probability .
Appendix C Proof of Theorem 1
We define and . We also define linear operators , and by , and . Note are all self-adjoint. We denote the identity map by . The summations indicate the summation over . The maximum indicate the maximum over