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+---
+layout: post
+title: Performance Optimizations: ComplEx interaction function 
+author: Max Berrendorf
+date: 2020-12-?? 00:00:00 -0800
+---
+Providing performant code is one of the main objectives of the pykeen development team:
+Having performant code allows users to faster train and evaluate models, thus accelerating model search and inference.
+It also improves resource utilization and since it likely avoids unnecessary computations, also improves environmental impact.
+
+The computational bottleneck of training and evaluating Knowledge Graph Embedding Models (KGEM) usually lies in their interaction function:
+Whenever, we compute scores for a triple we need to evaluate it.
+<!-- TODO: add trace for training loop -->
+Thus, optimizing the interaction function's implementation is the primary target for optimizing overall runtime.
+
+Before we dive into the optimization let us first revisit the use cases within a link prediction pipeline.
+* SLCWA: 1:1 / 1:k
+* LCWA / Evaluation: 1:n
+
+
+We can generalize the score function to support all the aforementioned use cases (and even a few more, like full CWA training) to the general form of the interaction function
+```python
+import torch
+
+def score(h: torch.FloatTensor, r: torch.FloatTensor, t: torch.FloatTensor):
+    """
+    :param h: shape: (batch_size, num_heads, dim)
+        The head representations.
+    :param r: shape: (batch_size, num_relations, dim)
+        The relation representations.
+    :param t: shape: (batch_size, num_tails, dim)
+        The tail representations.
+        
+    :return: shape: (batch_size, num_heads, num_relations, num_tails)
+        The scores.
+    """
+```
+with the following input shapes for the individual use cases 
+| Use Case | h | r | t | 
+| -- | -- | -- | -- |
+| score_hrt (naive, 1:1) | (b * (s + 1), 1, d) | (b * (s + 1), 1, d) | (b * (s + 1), 1, d) |
+| score_hrt (fast h, 1:k) | (b, s + 1, d) | (b, 1, d) | (b, 1, d) |
+| score_hrt (fast t, 1:k) | (b, 1, d) | (b, 1, d) | (b, s + 1, d) |
+| score_h (1:n) | (1, n, d) | (b, 1, d) | (b, 1, d) |
+| score_t (1:n) | (b, 1, d) | (b, 1, d) | (1, n, d) |
+
+In this blog post, we will exemplify how we optimize interaction function implementation at the example of ComplEx.
+For head, relation and tail representations $h, r, t \in \mathbb{C}^d$, the ComplEx interaction function is given by 
+```math
+score(h, r, t) = Re(<h, r, conj(t)>)
+               = <Re(h), Re(r), Re(t)> - <Re(h), Im(r), -Im(t)> - <Im(h), Re(r), -Im(t)>  - <Im(h), Im(r), Re(t)>
+               = <Re(h), Re(r), Re(t)> + <Re(h), Im(r), Im(t)> + <Im(h), Re(r), Im(t)>  - <Im(h), Im(r), Re(t)>
+```
+
+Notably, we can write
+```math
+score(h, r, t) = Re(<h \odot r, conj(t)>) = Re(<h, r \odot conj(t)>) 
+```
+* expected cost (hr)t / h(rt)
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