Bring Your Own Interaction
==========================

This is a tutorial about how to implement your own interaction modules (also known as scoring functions) as subclasses
of :class:`pykeen.nn.modules.Interaction` for use in PyKEEN.

Implementing your first Interaction Module
------------------------------------------

Imagine you've taken a time machine back to 2013 and you have just invented TransE, defined as:

.. math::

    f(h, r, t) = -\| \mathbf{e}_h + \mathbf{r}_r - \mathbf{e}_t \|_2

where $\mathbf{e}_i$ is the $d$-dimensional representation for entity $i$, $\mathbf{r}_j$ is the $d$-dimensional
representation for relation $j$, and $\|...\|_2$ is the $L_2$ norm.

To implement TransE in PyKEEN, you need to subclass the :class:`pykeen.nn.modules.Interaction`. This class it itself a
subclass of :class:`torch.nn.Module`, which means that you need to provide an implementation of
:meth:`torch.nn.Module.forward`. However, the arguments are predefined as ``h``, ``r``, and ``t``, which correspond to
the representations of the head, relation, and tail, respectively.

.. code-block:: python

    from pykeen.nn.modules import Interaction


    class TransEInteraction(Interaction):
        def forward(self, h, r, t):
            return -(h + r - t).norm(p=2, dim=-1)

Note the ``dim=-1`` because this operation is actually defined over an entire batch of head, relation, and tail
representations.

.. seealso::

    A reference implementation is provided in :class:`pykeen.nn.modules.TransEInteraction`

As a researcher who just invented TransE, you might wonder what would happen if you replaced the addition ``+`` with
multiplication ``*``. You might then end up with a new interaction like this (which just happens to be DistMult, which
was published just a year after TransE):

.. math::

    f(h, r, t) = \mathbf{e}_h^T diag(\mathbf{r}_r) \mathbf{e}_t

where $\mathbf{e}_i$ is the $d$-dimensional representation for entity $i$, $\mathbf{r}_j$ is the $d$-dimensional
representation for relation $j$.

.. code-block:: python

    from pykeen.nn.modules import Interaction


    class DistMultInteraction(Interaction):
        def forward(self, h, r, t):
            return (h * r * t).sum(dim=-1)

.. seealso::

    A reference implementation is provided in :class:`pykeen.nn.modules.DistMultInteraction`

Interactions with Hyper-Parameters
----------------------------------

While we previously defined TransE with the $L_2$ norm, it could be calculated with a different value for $p$:

.. math::

    f(h, r, t) = -\| \mathbf{e}_h + \mathbf{r}_r - \mathbf{e}_t \|_p

This could be incorporated into the interaction definition by using the ``__init__()``, storing the value for $p$ in the
instance, then accessing it in ``forward()``.

.. code-block:: python

    from pykeen.nn.modules import Interaction


    class TransEInteraction(Interaction):
        def __init__(self, p: int):
            super().__init__()
            self.p = p

        def forward(self, h, r, t):
            return -(h + r - t).norm(p=self.p, dim=-1)

In general, you can put whatever you want in ``__init__()`` to support the calculation of scores.

Interactions with Trainable Parameters
--------------------------------------

In ER-MLP, the multi-layer perceptron consists of an input layer with $3 \times d$ neurons, a hidden layer with $d$
neurons and output layer with one neuron. The input is represented by the concatenation embeddings of the heads,
relations and tail embeddings. It is defined as:

.. math::

    f(h, r, t) = W_2 ReLU(W_1 cat(h, r, t) + b_1) + b_2

with hidden dimension $y$, $W_1 \in \mathcal{R}^{3d \times y}$, $W_2\in \mathcal{R}^y$, and biases $b_1 \in
\mathcal{R}^y$ and $b_2 \in \mathcal{R}$.

$W_1$, $W_1$, $b_1$, and $b_2$ are *global* parameters, meaning that they are trainable, but are neither attached to the
entities nor relations. Unlike the $p$ in TransE, these global trainable parameters are not considered hyper-parameters.
However, like hyper-parameters, they can also be defined in the `__init__` function of your
:class:`pykeen.nn.modules.Interaction` class. They are trained jointly with the entity and relation embeddings during
training.

.. code-block:: python

    import torch.nn
    from pykeen.nn.modules import Interaction
    from pykeen.utils import broadcast_cat


    class ERMLPInteraction(Interaction):
        def __init__(self, embedding_dim: int, hidden_dim: int):
            super().__init__()
            # The weights of this MLP will be learned.
            self.mlp = torch.nn.Sequential(
                torch.nn.Linear(in_features=3 * embedding_dim, out_features=hidden_dim, bias=True),
                torch.nn.ReLU(),
                torch.nn.Linear(in_features=hidden_dim, out_features=1, bias=True),
            )

        def forward(self, h, r, t):
            x = broadcast_cat([h, r, t], dim=-1)
            return self.mlp(x)

Note that :func:`pykeen.utils.broadcast_cat` was used instead of the standard :func:`torch.cat` because of the
standardization of shapes of head, relation, and tail vectors.

.. seealso::

    A reference implementation is provided in :class:`pykeen.nn.modules.ERMLPInteraction`

Interactions with Different Shaped Vectors
------------------------------------------

The Structured Embedding uses a 2-tensor for representing each relation, with an interaction defined as:

.. math::

    f(h, r, t) = - \|\textbf{M}_{r}^{head} \textbf{e}_h  - \textbf{M}_{r}^{tail} \textbf{e}_t\|_p

where $\mathbf{e}_i$ is the $d$-dimensional representation for entity $i$, $\mathbf{M}^{head}_j$ is the $d \times
d$-dimensional representation for relation $j$ for head entities, $\mathbf{M}^{tail}_j$ is the $d \times d$-dimensional
representation for relation $j$ for tail entities, and $\|...\|_2$ is the $L_p$ norm.

For the purposes of this tutorial, we will propose a simplification to Strucuterd Embedding (also similar to TransR)
where the same relation 2-tensor is used to project both the head and tail entities as in:

.. math::

    f(h, r, t) = - \|\textbf{M}_{r} \textbf{e}_h  - \textbf{M}_{r} \textbf{e}_t\|_2

where $\mathbf{e}_i$ is the $d$-dimensional representation for entity $i$, $\mathbf{M}_j$ is the $d \times
d$-dimensional representation for relation $j$, and $\|...\|_2$ is the $L_2$ norm.

.. code-block:: python

    from pykeen.nn.modules import Interaction


    class SimplifiedStructuredEmbeddingInteraction(Interaction):
        relation_shape = ("dd",)

        def forward(self, h, r, t):
            h_proj = r @ h.unsqueeze(dim=-1)
            t_proj = r @ t.unsqueeze(dim=-1)
            return -(h_proj - t_proj).squeeze(dim=-1).norm(p=2, dim=-1)

Note the definition of the ``relation_shape``. By default, the ``entity_shape`` and ``relation_shape`` are both equal to
``('d', )``, which uses eigen-notation to show that they both are 1-tensors with the same shape. In this simplified
version of Structured Embedding, we need to denote that the shape of the relation is $d \times d$, so it's written as
``dd``.

.. seealso::

    Reference implementations are provided in :class:`pykeen.nn.modules.StructuredEmbeddingInteraction` and in
    :class:`pykeen.nn.modules.TransRInteraction`.

Interactions with Multiple Representations
------------------------------------------

Sometimes, like in the canonical version of Structured Embedding, you need more than one representation for entities
and/or relations. To specify this, you just need to extend the tuple for ``relation_shape`` with more entries, each
corresponding to the sequence of representations.

.. code-block:: python

    from pykeen.nn.modules import Interaction


    class StructuredEmbeddingInteraction(Interaction):
        relation_shape = (
            "dd",  # Corresponds to $\mathbf{M}^{head}_j$
            "dd",  # Corresponds to $\mathbf{M}^{tail}_j$
        )

        def forward(self, h, r, t):
            # Since the relation_shape is more than length 1, the r value is given as a sequence
            # of the representations defined there. You can use tuple unpacking to get them out
            r_h, r_t = r
            h_proj = r_h @ h.unsqueeze(dim=-1)
            t_proj = r_t @ t.unsqueeze(dim=-1)
            return -(h_proj - t_proj).squeeze(dim=-1).norm(p=2, dim=-1)

Interactions with Different Dimension Vectors
---------------------------------------------

TransD is an example of an interaction module that not only uses two different representations for each entity and two
representations for each relation, but they are of different dimensions.

It can be implemented by choosing a different letter for use in the ``entity_shape`` and/or ``relation_shape``
dictionary. Ultimately, the letters used are arbitrary, but you need to remember what they are when using the
:func:`pykeen.models.make_model`, :func:`pykeen.models.make_model_cls`, or :func:`pykeen.pipeline.interaction_pipeline`
functions to instantiate a model, make a model class, or run the pipeline using your custom interaction module
(respectively).

.. code-block:: python

    from pykeen.nn.modules import Interaction
    from pykeen.utils import project_entity


    class TransDInteraction(Interaction):
        entity_shape = ("d", "d")
        relation_shape = ("e", "e")

        def forward(self, h, r, t):
            h, h_proj = h
            r, r_proj = r
            t, t_proj = t
            h_bot = project_entity(
                e=h,
                e_p=h_p,
                r_p=r_p,
            )
            t_bot = project_entity(
                e=t,
                e_p=t_p,
                r_p=r_p,
            )
            return -(h_bot + r - t_bot).norm(p=2, dim=-1)

.. note::

    The :func:`pykeen.utils.project_entity` function was used in this implementation to reduce the complexity. So far,
    it's the case that all of the models using multiple different representation dimensions are quite complicated and
    don't fall into the paradigm of presenting simple examples.

.. seealso::

    A reference implementation is provided in :class:`pykeen.nn.modules.TransDInteraction`

Differences between :class:`pykeen.nn.modules.Interaction` and :class:`pykeen.models.Model`
-------------------------------------------------------------------------------------------

The high-level :func:`pipeline` function allows you to pass pre-defined subclasses of :class:`pykeen.models.Model` such
as :class:`pykeen.models.TransE` or :class:`pykeen.models.DistMult`. These classes are high-level wrappers around the
interaction functions :class:`pykeen.nn.modules.TransEInteraction` and :class:`nn.modules.DistMultInteraction` that are
more suited for running benchmarking experiments or practical applications of knowledge graph embeddings that include
lots of information about default hyper-parameters, recommended hyper-parameter optimization strategies, and more
complex applications of regularization schemas.

As a researcher, the :class:`pykeen.nn.modules.Interaction` is a way to quickly translate ideas into new models that can
be used without all of the overhead of defining a :class:`pykeen.models.Model`. These components are also completely
reusable throughout PyKEEN (e.g., in self-rolled training loops) and can be used as standalone components outside of
PyKEEN.

If you are happy with your interaction module and would like to go the next step to making it generally reusable, check
the "Extending the Models" tutorial.

*Ad hoc* Models from Interactions
---------------------------------

.. automodule:: pykeen.models.resolve

Interaction Pipeline
--------------------

The :func:`pykeen.pipeline.pipeline` also allows passing of an interaction such that the following code block can be
compressed:

.. code-block:: python

    from pykeen.pipeline import pipeline
    from pykeen.nn.modules import TransEInteraction

    model = make_model_cls(
        interaction=TransEInteraction,
        interaction_kwargs={"p": 2},
        dimensions={"d": 100},
    )
    results = pipeline(dataset="Nations", model=model, ...)

into:

.. code-block:: python

    from pykeen.pipeline import pipeline
    from pykeen.nn.modules import TransEInteraction

    results = pipeline(
        dataset="Nations", interaction=TransEInteraction, interaction_kwargs={"p": 2}, dimensions={"d": 100}, ...
    )

This can be used with any subclass of the :class:`pykeen.nn.modules.Interaction`, not only ones that are implemented in
the PyKEEN package.