A Three-Way Model for Collective Learning on Multi-Relational Data

Maximilian NickelVolker TrespHans‐Peter Kriegel

article2011ICML2,540 citations

Introduces RESCAL, a tensor factorization model for multi-relational data that enables collective learning across interconnected entities and asymmetric relations with an efficient alternating least-squares algorithm that scales significantly faster than standard relational learning methods.

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Relational data from networks and the semantic web is growing rapidly in areas such as social modeling and bioinformatics, yet standard methods struggle to capture correlations across interconnected entities. This limits performance on tasks like link prediction and entity resolution where collective learning from related items can improve accuracy.

The article set out to develop and test a tensor factorization approach called RESCAL that models multi-relational data while enabling collective learning through shared latent components.

The authors represent relations as a three-way tensor and factorize each slice as Xk ≈ A Rk A^T. They solve the model with an alternating least-squares algorithm adapted from DEDICOM methods and evaluate it on a new US presidents dataset plus established benchmarks including Cora, Kinships, Nations, and UMLS. Experiments use cross-validation and compare against CP, DEDICOM, BCTF, IRM, MRC, and SUNS variants.

RESCAL achieved AUC scores comparable or superior to state-of-the-art methods on link prediction and entity resolution while running in minutes rather than hours. It clearly outperformed CP and non-aggregated baselines on collective tasks, with gains of 0.3–0.5 AUC points in the presidents example. Runtime was often an order of magnitude faster than DEDICOM and competitive with CP despite the richer model. The method also produced meaningful clusters of countries in the Nations data.

These results indicate that RESCAL can deliver higher-quality predictions for relational tasks at lower computational cost, supporting faster decisions in domains that rely on network data. The shared latent space automatically propagates information across relations, removing the need for manual feature engineering.

The authors recommend exploring distributed and stochastic gradient versions for larger data and adding constraints such as typed relations to further improve speed and accuracy. Additional work on noisy relation aggregation would strengthen the approach before wide deployment.

The main limitations are the lack of rank reduction on the relation mode, which may hurt performance when relations are noisy, and slower scaling with model rank in regularized settings. Results rest on several standard benchmarks and controlled experiments, giving reasonable confidence for similar data but warranting caution on very large or sparse graphs without further validation.

Ludwig Maximilian University of Munich Siemens Corporate Research efficient algorithm factorization relational data models RESCAL Tensor Factorization Collective Learning

mnick/rescal.py Nickel et al (2011).pdf

Paper: The link prediction problem for social networks, David Liben-Nowell et al. (2003). This foundational paper establishes the link prediction formulation and baseline structural heuristics that multi-relational tensor models aim to generalize.
Paper: Factorization Machines, Steffen Rendle (2010). This work introduces low-rank parameter factorizations to capture higher-order variable interactions in sparse relational data.
Paper: Probabilistic Matrix Factorization, Andriy Mnih et al. (2007). It provides the low-rank matrix decomposition principles and latent factor modeling foundations that RESCAL extends to multi-relational tensor slices.
Paper: Markov logic networks, Matthew Richardson et al. (2006). It presents the core collective learning problem and probabilistic reasoning framework over multi-relational knowledge domains.

Paper: Complex Embeddings for Simple Link Prediction, Théo Trouillon et al. (2016). This work directly compares against RESCAL while introducing complex embeddings to capture symmetric and antisymmetric multi-relational patterns efficiently.
Paper: Embedding Entities and Relations for Learning and Inference in Knowledge Bases, Bishan Yang et al. (2014). It builds upon multi-relational tensor factorization like RESCAL to establish a unified linear and bilinear embedding framework for knowledge base completion.
Paper: Reasoning With Neural Tensor Networks for Knowledge Base Completion, Richard Socher et al. (2013). It extends bilinear multi-relational tensor modeling into deep neural tensor architectures for relational reasoning.
Paper: Translating Embeddings for Modeling Multi-relational Data, Antoine Bordes et al. (2013). It proposes translational distance models as a highly scalable alternative to full bilinear tensor factorizations for multi-relational knowledge graph completion.
Paper: Knowledge Graph Embedding by Translating on Hyperplanes, Zhen Wang et al. (2014). It extends relational embedding methods by projecting entities onto relation-specific hyperplanes to better capture multi-relational mapping properties.
Paper: Learning Entity and Relation Embeddings for Knowledge Graph Completion, Yankai Lin et al. (2015). It further advances relational representation learning by mapping entities into separate relation-specific vector spaces.
Paper: RotatE: Knowledge Graph Embedding by Relational Rotation in Complex Space, Zhiqing Sun et al. (2019). It generalizes multi-relational link prediction by modeling relations as rotations in complex space to simultaneously represent symmetry, antisymmetry, and composition.
Paper: Modeling Relational Data with Graph Convolutional Networks, Michael Schlichtkrull et al. (2018). It adopts relational graph convolutional networks as neural encoders paired with tensor-style decoders to model multi-relational knowledge graphs.
Paper: Convolutional 2D Knowledge Graph Embeddings, Tim Dettmers et al. (2017). It advances knowledge graph link prediction beyond shallow bilinear factorizations by applying 2D convolutions over entity and relation embeddings.

Cover for A Three-Way Model for Collective Learning on Multi-Relational Data

Abstract

Relational learning is becoming increasingly important in many areas of application. Here, we present a novel approach to relational learning based on the factorization of a three-way tensor. We show that unlike other tensor approaches, our method is able to perform collective learning via the latent components of the model and provide an efficient algorithm to compute the factorization. We substantiate our theoretical considerations regarding the collective learning capabilities of our model by the means of experiments on both a new dataset and a dataset commonly used in entity resolution. Furthermore, we show on common benchmark datasets that our approach achieves better or on-par results, if compared to current state-of-the-art relational learning solutions, while it is significantly faster to compute.

Topics

efficient algorithm factorization relational data models relational learning

Keywords

RESCAL Tensor Factorization Collective Learning Multi-Relational Data Alternating Least Squares CP DEDICOM Link Prediction Entity Resolution Runtime Performance

1. Introduction
2. Modelling and Notation
3. Related Work
4. Methods and Theoretical Aspects
4.1. A Model for Multi-Relational Data
4.2. Connections to other Tensor Factorizations
4.3. Computing the Factorization
4.4. Solving Relational Learning Tasks
5. Evaluation
5.1. Collective Classification
5.2. Collective Entity Resolution
5.3. Kinships, Nations and UMLS
5.4. Runtime Performance and Technical Considerations
6. Conclusion and Future Work
Acknowledgements
References

Knowls

Knowl 1 — RESCAL Tensor Factorization Model
model/method
RESCAL is a tensor factorization method for multi-relational data. Relational data with $n$ entities and $m$ relation types is represented as a three-way binary tensor $\mathcal{X} \in \mathbb{R}^{n \times n \times m}$ , where $\mathcal{X}_{ijk} = 1$ if the triple $(i\text{-th entity}, k\text{-th relation}, j\text{-th entity})$ exists and $0$ otherwise.

RESCAL factorizes each $n \times n$ frontal slice $X_k$ of $\mathcal{X}$ into a rank- $r$ decomposition: $X_k \approx A R_k A^T, \quad \text{for } k = 1, \dots, m$ where:
- $A \in \mathbb{R}^{n \times r}$ is a shared factor matrix containing the $r$ -dimensional latent-component representations of all $n$ entities across all relations, serving simultaneously as subject and object representations.
- $R_k \in \mathbb{R}^{r \times r}$ is an asymmetric relation-specific matrix modeling interactions between the latent components for the $k$ -th relation type.
The factor matrices $A$ and $\{R_k\}_{k=1}^m$ are computed by minimizing the regularized squared loss: $\min_{A, \{R_k\}} f(A, R_k) + g(A, R_k)$ with $f(A, R_k) = \frac{1}{2} \sum_{k=1}^m \|X_k - A R_k A^T\|_F^2 = \frac{1}{2} \sum_{i=1}^n \sum_{j=1}^n \sum_{k=1}^m (X_{ijk} - \mathbf{a}_i^T R_k \mathbf{a}_j)^2$ $g(A, R_k) = \frac{1}{2} \lambda \left( \|A\|_F^2 + \sum_{k=1}^m \|R_k\|_F^2 \right)$ where $\mathbf{a}_i, \mathbf{a}_j \in \mathbb{R}^r$ denote the $i$ -th and $j$ -th rows of $A$ , $\|\cdot\|_F$ is the Frobenius norm, and $\lambda \ge 0$ is a regularization parameter.
Knowl 2 — Alternating Least Squares Optimization for RESCAL
algorithm
The RESCAL optimization problem is solved via an Alternating Least Squares (ALS) procedure that alternates between updating the entity representation matrix $A \in \mathbb{R}^{n \times r}$ and the relation interaction matrices $R_k \in \mathbb{R}^{r \times r}$ for $k = 1, \dots, m$ until relative reconstruction error converges.
Input: Frontal slices $X_1, \dots, X_m \in \mathbb{R}^{n \times n}$ , rank $r$ , regularization parameter $\lambda$ , convergence threshold $\epsilon$ , maximum iterations $T_{max}$
Output: Entity matrix $A \in \mathbb{R}^{n \times r}$ , relation matrices $R_1, \dots, R_m \in \mathbb{R}^{r \times r}$
Initialize $A$ from the leading $r$ eigenvectors of $\sum_{k=1}^m (X_k + X_k^T)$ (or randomly)
Initialize $R_1, \dots, R_m$ randomly
repeat
    Compute $\Gamma \leftarrow \sum_{k=1}^m (X_k A R_k^T + X_k^T A R_k)$
    Compute $\Psi \leftarrow \sum_{k=1}^m (R_k A^T A R_k^T + R_k^T A^T A R_k) + \lambda I_r$
     $A \leftarrow \Gamma \Psi^{-1}$
     $Z \leftarrow A \otimes A$
    for $k \leftarrow 1$ to $m$ do
         $\text{vec}(R_k) \leftarrow (Z^T Z + \lambda I_{r^2})^{-1} Z^T \text{vec}(X_k)$
    end for
until $\frac{\sum_{k=1}^m \|X_k - A R_k A^T\|_F^2}{\|\mathcal{X}\|_F^2}$ converges within $\epsilon$ or iteration limit $T_{max}$ reached
return $A, R_1, \dots, R_m$
The update for $A$ is obtained by stacking frontal slices as $\bar{X} = (X_1, X_1^T, \dots, X_m, X_m^T)$ and solving for the left factor while holding the right factor fixed to obtain a closed-form normal equations step. For unregularized $R_k$ ( $\lambda = 0$ ), $(Z^T Z)^{-1} Z^T = ((A^T A)^{-1} A^T) \otimes ((A^T A)^{-1} A^T)$ , reducing the Kronecker inversion to inverting the smaller $r \times r$ matrix $A^T A$ . QR decomposition of $A$ and $X_k$ can make updates of $R_k$ independent of entity count $n$ .
Knowl 3 — Information Propagation and Collective Learning Mechanism in RESCAL
theoretical result
Collective learning refers to exploiting the attributes, relations, and classes of interconnected entities to improve prediction accuracy across a relational network.

In RESCAL, the reconstruction of the relation between entity $i$ and entity $j$ under relation $k$ is: $\hat{X}_{ijk} = \mathbf{a}_i^T R_k \mathbf{a}_j$ where $\mathbf{a}_i, \mathbf{a}_j \in \mathbb{R}^r$ are the $i$ -th and $j$ -th rows of $A$ .

Because each entity has a single unique latent-component representation $\mathbf{a}_i$ regardless of whether it acts as a subject or an object:
1. The calculation of $\mathbf{a}_i$ directly depends on the latent representations $\mathbf{a}_j$ of all entities connected to $i$ across all relations.
2. In turn, each $\mathbf{a}_j$ depends on entities connected to $j$ . Consequently, information propagates transitively through the latent representations along paths of arbitrary length in the relational graph.
3. The asymmetric relation-specific matrix $R_k \in \mathbb{R}^{r \times r}$ captures how the latent components interact specifically for relation $k$ , preserving directional distinctions between subject and object roles.
In contrast, standard CANDECOMP/PARAFAC (CP) assigns separate factor matrices to subject and object modes, breaking this bidirectional information flow mechanism.
Knowl 4 — Connections of RESCAL to DEDICOM and Tucker3 Decompositions
theoretical result
The RESCAL model ( $X_k \approx A R_k A^T$ ) connects to standard tensor decomposition models in psycho- and chemometrics:
1. Relaxation of DEDICOM: Rank- $r$ DEDICOM factorizes three-way slices as $X_k \approx A D_k R D_k A^T$ , where $A \in \mathbb{R}^{n \times r}$ contains entity components, $R \in \mathbb{R}^{r \times r}$ is a single global interaction matrix, and $D_k \in \mathbb{R}^{r \times r}$ is a diagonal matrix weighting latent component participation in relation $k$ . RESCAL relaxes DEDICOM by replacing $D_k R D_k$ with an unconstrained relation-specific interaction matrix $R_k \in \mathbb{R}^{r \times r}$ .
2. Extension of IDIOSCAL: RESCAL is an asymmetric extension of IDIOSCAL, which models symmetric slices as $X_k \approx A R_k A^T$ with symmetric $R_k$ .
3. Restricted Tucker3 Model: A standard Tucker3 decomposition represents a matricized tensor as $X_{(n)} = A G_{(n)} (C \otimes B)^T$ . RESCAL is equivalent to Tucker3 with constraints $B = A$ , $C = I_m$ , and core tensor slices $G_k = R_k$ .
Knowl 5 — Relational Learning Task Formulations in RESCAL
model/method
Using the learned parameters $A \in \mathbb{R}^{n \times r}$ and $R_k \in \mathbb{R}^{r \times r}$ , RESCAL addresses multiple relational learning tasks:
1. Link Prediction: The score of a triple $(i, k, j)$ is computed from the rank-reduced slice reconstruction $\hat{X}_{ijk} = \mathbf{a}_i^T R_k \mathbf{a}_j$ . Triples are classified by thresholding $\hat{X}_{ijk} > \theta$ or ranked by score.
2. Collective Classification: Entity classes are represented by creating dedicated class-membership relations in $\mathcal{X}$ and adding class labels as entities. Classification is solved via link prediction on the corresponding class-relation slice.
3. Collective Entity Resolution: The likelihood that entity $x$ and entity $y$ refer to the same entity is computed via a heat kernel on their normalized latent vectors in $A$ : $k(x, y) = \exp\left( -\frac{\|\tilde{\mathbf{a}}_x - \tilde{\mathbf{a}}_y\|_2^2}{\delta} \right)$ where $\tilde{\mathbf{a}}_x = \mathbf{a}_x / \|\mathbf{a}_x\|_2$ and $\delta > 0$ is a user-defined scaling parameter.
4. Link-Based Clustering: Relational clustering is performed by applying clustering algorithms (such as $k$ -means) directly to the entity rows of $A$ .
Knowl 6 — Link Prediction Benchmark Results on Kinships, Nations, and UMLS
empirical result
RESCAL was evaluated against tensor and relational learning baselines on 10-fold cross-validation link prediction using $(subject, predicate, object)$ triples as the statistical unit. Performance was measured by Area Under the Precision-Recall Curve (AUC-PR):
- Kinships ( $n=104$ entities, $m=26$ relations):
  - RESCAL: $0.95$
  - CP: $0.94$
  - BCTF: $0.90$
  - MRC: $0.85$
  - DEDICOM: $0.69$
  - IRM: $0.66$
- UMLS ( $n=135$ entities, $m=49$ relations):
  - RESCAL: $0.98$
  - BCTF: $0.98$
  - MRC: $0.98$
  - CP: $0.95$
  - DEDICOM: $0.95$
  - IRM: $0.70$
- Nations ( $n=125$ entities, $m=57$ relations):
  - RESCAL: $0.84$
  - CP: $0.83$
  - DEDICOM: $0.81$
  - IRM: $0.75$
  - MRC: $0.75$
RESCAL achieved performance superior or comparable to specialized Bayesian and Markov logic relational models (BCTF, IRM, MRC) across all three benchmarks.
Knowl 7 — Collective Entity Resolution Results on the Cora Dataset
empirical result
RESCAL was evaluated on 5-fold cross-validation entity resolution on the Cora dataset represented as a $2497 \times 2497 \times 7$ tensor. Entity similarity was computed via heat kernel similarity in the normalized latent space $A$ . Performance was measured by Area Under the Precision-Recall Curve (AUC-PR) across three entity types:
- Citations (central node directly connected to features):
  - MLN(BCTS): $0.988$
  - MLN(B): $0.915$
  - RESCAL: $0.896$
  - CP: $0.886$
- Author (requires collective propagation through citations):
  - MLN(BCTS): $0.992$
  - MLN(B): $0.987$
  - RESCAL: $0.970$
  - CP: $0.601$
- Venue (requires collective propagation through citations):
  - RESCAL: $0.911$
  - MLN(BCTS): $0.807$
  - MLN(B): $0.736$
  - CP: $0.387$
While CP performs similarly to RESCAL on citations ( $0.886$ vs $0.896$ ), it fails on authors ( $0.601$ ) and venues ( $0.387$ ). RESCAL effectively leverages collective information from connected citation entities, outperforming Markov Logic Networks on venue resolution.
Knowl 8 — Collective Classification on US Presidents Dataset
empirical result
RESCAL was evaluated on a collective classification task on a DBpedia US presidents dataset ( $93 \times 93 \times 3$ tensor with relations presidentOf, vicePresidentOf, and party). The task is to predict the masked political party of persons in 10-fold cross-validation.

Area Under the Precision-Recall Curve (AUC-PR) results:
- RESCAL: $0.78$
- SUNS+AG (SUNS with manual feature aggregation): $0.74$
- DEDICOM: $0.64$
- SUNS (relational regression without aggregation): $0.48$
- CP: $0.44$
- Random baseline: $0.16$
RESCAL achieved the highest AUC ( $0.78$ ), demonstrating its ability to propagate party affiliation between related presidents and vice presidents automatically. CP performed similarly to unaggregated SUNS ( $0.44$ vs $0.48$ ), demonstrating that standard CP cannot perform collective learning.

Knowl 9 — Runtime Comparison of RESCAL, CP-ALS, and ASALSAN

data/table

The runtime of regularized RESCAL was evaluated against CP-ALS and ASALSAN (DEDICOM) on four datasets across decomposition ranks $r \in \{10, 20, 40\}$ on an Intel Core 2 Duo 2.5 GHz system with convergence threshold $\epsilon = 10^{-5}$ .

Dataset	Algorithm	Total Runtime
		Rank 10	Rank 20	Rank 40
Kinships	CP-ALS	6.4s	25.4s	105.8s
$\|E\|: 104, \|R\|: 26$	ASALSAN	527s	1549s	16851s
	RESCAL	1.1s	3.7s	51.2s
Nations	CP-ALS	16.4s	43.8s	68.3s
$\|E\|: 125, \|R\|: 57$	ASALSAN	830s	4602s	42506s
	RESCAL	1.7s	5.3s	54.4s
UMLS	CP-ALS	5.5s	11.7s	53.9s
$\|E\|: 135, \|R\|: 49$	ASALSAN	1706s	4846s	6012s
	RESCAL	2.6s	4.9s	72.3s
Cora	CP-ALS	369s	934s	3190s
$\|E\|: 2497, \|R\|: 7$	ASALSAN	132s	154s	–
	RESCAL	364s	348s	680s

$|E|$ is the entity count, $|R|$ is the relation count, and -- indicates non-convergence. RESCAL is significantly faster than ASALSAN because it eliminates DEDICOM's diagonal factor updates and shares $(Z^T Z)^{-1}$ across relations. On relational datasets RESCAL also converged faster than CP-ALS (e.g., 348s vs 934s on Cora at rank 20), while IRM and MRC required over 10 hours per fold on similar datasets.

Knowl 10 — Uncompressed Relation Mode and Rank Inversion Scaling Limitations
limitation
RESCAL has two main structural and computational limitations:
1. Uncompressed Relation Dimension: In its equivalent Tucker3 formulation ( $X_{(n)} = A G_{(n)} (C \otimes B)^T$ with $B = A$ and $C = I_m$ ), RESCAL applies rank reduction only to the entity mode via $A \in \mathbb{R}^{n \times r}$ , leaving the relation mode unreduced ( $C = I_m$ ). As a result, when relation types are noisy or redundant, RESCAL cannot compress or aggregate along the relation dimension simultaneously with entities.
2. Complexity Scaling with Rank: Updating regularized relation matrices $R_k \in \mathbb{R}^{r \times r}$ requires inverting $(Z^T Z + \lambda I_{r^2})$ , where $Z = A \otimes A$ . Inverting this $r^2 \times r^2$ matrix has computational complexity $\mathcal{O}(r^6)$ , which leads to poorer runtime scaling with higher ranks $r$ compared to CP-ALS.

Coverage note — None was omitted; all primary contributions—the RESCAL model formulation, ALS algorithm, theoretical relations to DEDICOM/Tucker3, relational task reductions, empirical benchmarks (US Presidents, Cora, Kinships/Nations/UMLS), runtime performance data, and stated model limitations—are captured.

References

1.Bader, Brett W. and Kolda, Tamara G. MATLAB tensor toolbox version 2.4, March 2010. URL http://csmr.ca.sandia.gov/~tgkolda/TensorToolbox/.
2.Bader, Brett W., Harshman, Richard A., and Kolda, Tamara G. Temporal analysis of semantic graphs using ASALSAN. In Seventh IEEE International Conference on Data Mining (ICDM 2007), pp. 33–42, Omaha, NE, USA, 2007. doi: 10.1109/ICDM.2007.54. URL http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=4470227.
3.Franz, T., Schultz, A., Sizov, S., and Staab, S. Triplerank: Ranking semantic web data by tensor decomposition. The Semantic Web-ISWC 2009, pp. 213–228, 2009.
4.Friedman, N., Getoor, L., Koller, D., and Pfeffer, A. Learning probabilistic relational models. In International Joint Conference on Artificial Intelligence, volume 16, pp. 1300–1309, 1999.
5.Getoor, L. and Taskar, B. Introduction to statistical relational learning. The MIT Press, 2007. ISBN 0262072882.
6.Harshman, R. A. Models for analysis of asymmetrical relationships among n objects or stimuli. In First Joint Meeting of the Psychometric Society and the Society for Mathematical Psychology, McMaster University, Hamilton, Ontario, August, 1978.
7.Harshman, R. A and Lundy, M. E. PARAFAC: parallel factor analysis. Computational Statistics & Data Analysis, 18(1): 39–72, 1994. ISSN 0167-9473.
8.Horn, Roger A. and Johnson, Charles R. Topics in matrix analysis. Cambridge University Press, June 1994. ISBN 9780521467131.
9.Huang, Yi, Tresp, Volker, Bundschus, Markus, and Rettinger, Achim. Multivariate structured prediction for learning on semantic web. In Proceedings of the 20th International Conference on Inductive Logic Programming (ILP), 2010.
10.Kemp, C., Tenenbaum, J. B, Griffiths, T. L, Yamada, T., and Ueda, N. Learning systems of concepts with an infinite relational model. In Proceedings of the National Conference on Artificial Intelligence, volume 21, pp. 381, 2006.
11.Kok, S. and Domingos, P. Statistical predicate invention. In Proceedings of the 24th international conference on Machine learning, pp. 433–440, 2007.
12.Kolda, Tamara G. and Bader, Brett W. Tensor decompositions and applications. SIAM Review, 51(3): 455—500, 2009. ISSN 00361445. doi: 10.1137/07070111X. URL http://link.aip.org/link/SIREAD/v51/i3/p455/s1&Agg=doi.
13.Richardson, M. and Domingos, P. Markov logic networks. Machine Learning, 62(1):107–136, 2006. ISSN 0885-6125.
14.Sen, P., Namata, G., Bilgic, M., Getoor, L., Galligher, B., and Eliassi-Rad, T. Collective classification in network data. AI Magazine, 29(3):93, 2008. ISSN 0738-4602.
15.Singh, A. P and Gordon, G. J. Relational learning via collective matrix factorization. In Proceeding of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 650–658, 2008.
16.Singla, P. and Domingos, P. Entity resolution with markov logic. In Data Mining, 2006. ICDM’06. Sixth International Conference on, pp. 572–582, 2007.
17.Sun, J., Tao, D., and Faloutsos, C. Beyond streams and graphs: dynamic tensor analysis. In Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 374–383, 2006. ISBN 1595933395.
18.Sutskever, I., Salakhutdinov, R., and Tenenbaum, J. B. Modelling relational data using bayesian clustered tensor factorization. Advances in Neural Information Processing Systems, 22, 2009.
19.Tucker, L. R. Some mathematical notes on three-mode factor analysis. Psychometrika, 31(3):279–311, 1966. ISSN 0033-3123.
20.Xu, Z., Tresp, V., Yu, K., and Kriegel, H. P. Infinite hidden relational models. In Proceedings of the 22nd International Conference on Uncertainty in Artificial Intelligence, 2006.

Citation

MLA

Nickel, M., et al. “A Three-Way Model for Collective Learning on Multi-Relational Data”. International Conference on Machine Learning, 2011, pp. 809–16, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.231.6909.

APA

Nickel, M., Tresp, V., & Kriegel, H. (2011). A Three-Way Model for Collective Learning on Multi-Relational Data. International Conference on Machine Learning, 809–816. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.231.6909

Chicago

Nickel, M., V. Tresp, and H. Kriegel. 2011. “A Three-Way Model for Collective Learning on Multi-Relational Data”. International Conference on Machine Learning, 809–16. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.231.6909.

Harvard

Nickel, M., Tresp, V. and Kriegel, H. (2011) “A Three-Way Model for Collective Learning on Multi-Relational Data”, International Conference on Machine Learning, pp. 809–816. Available at: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.231.6909.

Vancouver

1. Nickel M, Tresp V, Kriegel H (2011) A Three-Way Model for Collective Learning on Multi-Relational Data. International Conference on Machine Learning 809–816

BibTeX

@article{nickel2011three,
  title = {A Three-Way Model for Collective Learning on Multi-Relational Data},
  author = {Nickel, Maximilian and Tresp, Volker and Kriegel, Hans‐Peter},
  year = {2011},
  journal = {International Conference on Machine Learning},
  pages = {809-816},
  url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.231.6909}
}

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A Three-Way Model for Collective Learning on Multi-Relational Data

Abstract

Topics

Keywords

Table of Contents

Knowls

Knowl 1 — RESCAL Tensor Factorization Model

Knowl 2 — Alternating Least Squares Optimization for RESCAL

Knowl 3 — Information Propagation and Collective Learning Mechanism in RESCAL

Knowl 4 — Connections of RESCAL to DEDICOM and Tucker3 Decompositions

Knowl 5 — Relational Learning Task Formulations in RESCAL

Knowl 6 — Link Prediction Benchmark Results on Kinships, Nations, and UMLS

Knowl 7 — Collective Entity Resolution Results on the Cora Dataset

Knowl 8 — Collective Classification on US Presidents Dataset