Knowledge tracing, the act of modeling a student's knowledge through learning activities, is an extensively studied problem in the field of computer-aided education. Armed with attention mechanisms focusing on relevant information for target prediction, recurrent neural networks and Transformer-based knowledge tracing models have outperformed traditional approaches such as Bayesian knowledge tracing and collaborative filtering. However, the attention mechanisms of current state-of-the-art knowledge tracing models share two limitations. Firstly, the models fail to leverage deep self-attentive computations for knowledge tracing. As a result, they fail to capture complex relations among exercises and responses over time. Secondly, appropriate features for constructing queries, keys and values for the self-attention layer for knowledge tracing have not been extensively explored. The usual practice of using exercises and interactions (exercise-response pairs), as queries and keys/values, respectively, lacks empirical support.In this paper, we propose a novel Transformer-based model for knowledge tracing, SAINT: Separated Self-AttentIve Neural Knowledge Tracing. SAINT has an encoder-decoder structure where the exercise and response embedding sequences separately enter, respectively, the encoder and the decoder. The encoder applies self-attention layers to the sequence of exercise embeddings, and the decoder alternately applies selfattention layers and encoder-decoder attention layers to the sequence of response embeddings. This separation of input allows us to stack attention layers multiple times, resulting in an improvement in area under receiver operating characteristic curve (AUC). To the best of our knowledge, this is the first work to suggest an encoder-decoder model for knowledge tracing that applies deep self-attentive layers to exercises and responses separately.We empirically evaluate SAINT on a large-scale knowledge tracing dataset, EdNet, collected by an active mobile education
This paper proposes a deep learning-based denoising method for noisy low-dose computerized tomography (CT) images in the absence of paired training data. The proposed method uses a fidelity-embedded generative adversarial network (GAN) to learn a denoising function from unpaired training data of low-dose CT (LDCT) and standard-dose CT (SDCT) images, where the denoising function is the optimal generator in the GAN framework. This paper analyzes the f-GAN objective to derive a suitable generator that is optimized by minimizing a weighted sum of two losses: the Kullback-Leibler divergence between an SDCT data distribution and a generated distribution, and the 2 loss between the LDCT image and the corresponding generated images (or denoised image). The computed generator reflects the prior belief about SDCT data distribution through training. We observed that the proposed method allows the preservation of fine anomalous features while eliminating noise. The experimental results show that the proposed deep-learning method with unpaired datasets performs comparably to a method using paired datasets. A clinical experiment was also performed to show the validity of the proposed method for noise arising in the low-dose X-ray CT.
Advances in Artificial Intelligence in Education (AIEd) and the ever-growing scale of Interactive Educational Systems (IESs) have led to the rise of data-driven approaches for knowledge tracing and learning path recommendation. Unfortunately, collecting student interaction data is challenging and costly. As a result, there is no public largescale benchmark dataset reflecting the wide variety of student behaviors observed in modern IESs. Although several datasets, such as ASSISTments, Junyi Academy, Synthetic and STATICS are publicly available and widely used, they are not large enough to leverage the full potential of state-of-the-art data-driven models. Furthermore, the recorded behavior is limited to question-solving activities. To this end, we introduce EdNet, a large-scale hierarchical dataset of diverse student activities collected by Santa, a multi-platform self-study solution equipped with an artificial intelligence tutoring system. EdNet contains 131,417,236 interactions from 784,309 students collected over more than 2 years, making it the largest public IES dataset released to date. Unlike existing datasets, EdNet records a wide variety of student actions ranging from questionsolving to lecture consumption to item purchasing. Also, EdNet has a hierarchical structure which divides the student actions into 4 different levels of abstractions. The features of EdNet are domain-agnostic, allowing EdNet to be easily extended to different domains. The dataset is publicly released for research purposes. We plan to host challenges in multiple AIEd tasks with EdNet to provide a common ground for the fair comparison between different state-of-the-art models and to encourage the development of practical and effective methods.
Johnson recently proved Armstrong's conjecture which states that the average size of an (a, b)-core partition is (a+b+1)(a−1)(b−1)/24. He used various coordinate changes and one-to-one correspondences that are useful for counting problems about simultaneous core partitions. We give an expression for the number of (b 1 , b 2 , · · · , bn)-core partitions where {b 1 , b 2 , · · · , bn} contains at least one pair of relatively prime numbers. We also evaluate the largest size of a self-conjugate (s, s + 1, s + 2)-core partition. arXiv:1711.01469v1 [math.CO] 4 Nov 2017 2 JINEON BAEK, HAYAN NAM, AND MYUNGJUN YUArmstrong's conjecture by using Ehrhart theory. A proof without Ehrhart theory was given by Wang [13].In [8], Johnson estabilished a bijection between the set of (a, b)-cores and the setBy showing that the cardinality of this set is Cat a,b , he gave a new proof of Anderson's theorem. Inspired by Johnson's method and this bijection, we count the number of simultaneous core partitions. We find a general expression for the number of (b 1 , b 2 , . . . , b n )-core partitions where {b 1 , b 2 , . . . , b n } contains at least one pair of relatively prime numbers. As a corollary, we obtain an alternative proof for the number of (s, s + d, s + 2d)-core partitions, which was given by Yang-Zhong-Zhou [17] and Wang [13]. Subsequently, we also give a formula for the number of (s, s + d, s + 2d, s + 3d)-core partitions. Many authors have studied core partitions satisfying additional restrictions. For example, Berg and Vazirani [7] gave a formula for the number of a-core partitions with largest part x. We generalize this formula, giving a formula for the number of a-core partitions with largest part x and second largest part y.This paper also includes a result related to the largest size of a simultaneous core partition which has been studied by many mathematicians. For example, Aukerman, Kane and Sze [6, Conjecture 8.1] conjectured that if a and b are coprime, the largest size of an (a, b)-core partition is (a 2 − 1)(b 2 − 1)/24. This was proved by Tripathi in [12]. It is natural to wonder what would be the largest size of an (a, b, c)-core. Yang-Zhong-Zhou [17] found a formula for the largest size of an (s, s + 1, s + 2)-core. In section 4, we give a formula for the largest size of a selfconjugate (s, s + 1, s + 2)-core partition. We also prove that such a partition is unique (see Theorem 3.3).The layout of this paper is as follows. In Section 2, we introduce Johnson's ccoordinates and x-coordinates for core partitions. In Section 3, we give a formula for the largest size of a self-conjugate (s, s + 1, s + 2) core partition. In Section 4, using c-coordinates, we count the number of a-core partitions with given largest part and second largest part. In Section 5, we derive formulas for the number of simultaneous core partitions by using Johnson's z-coordinates.
Amdeberhan conjectured that the number of (s, s + 2)-core partitions with distinct parts for an odd integer s is 2 s−1 . This conjecture was first proved by Yan, Qin, Jin and Zhou, then subsequently by Zaleski and Zeilberger. Since the formula for the number of such core partitions is so simple one can hope for a bijective proof. We give the first direct bijective proof of this fact by establishing a bijection between the set of (s, s + 2)-core partitions with distinct parts and a set of lattice paths.
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