Metadata-Version: 2.4
Name: fxtensor_salmon
Version: 1.2.2
Summary: Markov Tensor Module
Author-email: Tadachika Oki <sakesnare@gmail.com>
License: Copyright 2025 Tadachika Oki
        
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Classifier: Programming Language :: Python :: 3
Classifier: License :: OSI Approved :: MIT License
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Classifier: Topic :: Scientific/Engineering :: Mathematics
Requires-Python: >=3.8
Description-Content-Type: text/markdown
License-File: LICENSE
Requires-Dist: numpy>=1.20.0
Provides-Extra: dev
Requires-Dist: pytest; extra == "dev"
Requires-Dist: ruff; extra == "dev"
Provides-Extra: discopy
Requires-Dist: discopy<2,>=1.2; extra == "discopy"
Dynamic: license-file

# FXTensor

FXTensor is a Python library for tensor-based computations, particularly suited for modeling probabilistic systems and processes inspired by category theory. It leverages NumPy for efficient numerical computations. The library primarily supports labeled indices for enhanced readability while maintaining compatibility with unlabeled numeric indices.

## Core Concepts

An FXTensor is defined by its `profile` and `data`, with optional string labels to make tensors more intuitive and meaningful.

- **Profile**: A pair `[domain, codomain]` specifying the dimensions of input (domain) and output (codomain) indices. For labeled tensors, e.g., `[[['a', 'b']], [['x', 'y', 'z']]]` represents a 2x3 matrix with labeled rows and columns. For unlabeled tensors, `[[2], [3]]` specifies dimensions numerically.
- **Labels (Optional)**: String labels can be assigned to each dimension, enhancing interpretability. For example, input axis labeled `['a', 'b']` and output axis labeled `['x', 'y', 'z']`. Unlabeled tensors have `labels` set to `None`.
- **Data**: A NumPy array holding the tensor’s values. Its shape must match the total number of dimensions in the profile (`len(domain) + len(codomain)`).

## Usage Examples

### Basic Example: Labeled Tensor

```python
import numpy as np
from fxtensor_salmon import FXTensor

# Create a 2x3 matrix with string labels
profile = [[['a', 'b']], [['x', 'y', 'z']]]
data = np.array([
    [0.1, 0.2, 0.7],  # a -> x, y, z
    [0.3, 0.3, 0.4]   # b -> x, y, z
])
tensor = FXTensor(profile, data=data)

# Access elements using labels
assert tensor.get_label_index(0, 'a') == 0  # Index of label 'a' on input axis
assert tensor.get_index_label(1, 2) == 'z'  # Label at index 2 on output axis
```

### Unlabeled Tensor

```python
# Create a 2x3 matrix with numeric indices
profile = [[2], [3]]
data = np.array([
    [0.1, 0.2, 0.7],
    [0.3, 0.3, 0.4]
])
tensor = FXTensor(profile, data=data)
assert tensor.labels == (None, None)  # No labels
```

### Creating Tensor from Strands

```python
# Create a tensor from labeled strands
profile = [[['a', 'b']], [['x', 'y', 'z']]]
strands = {
    "[[['a']], [['x']]]": 0.1,
    "[[['a']], [['y']]]": 0.2,
    "[[['a']], [['z']]]": 0.7,
    "[[['b']], [['x']]]": 0.3,
    "[[['b']], [['y']]]": 0.3,
    "[[['b']], [['z']]]": 0.4
}
tensor = FXTensor.from_strands(profile, strands)
assert tensor.labels == ([['a', 'b']], [['x', 'y', 'z']])
```

### Labeled Tensor Composition

```python
# P(Y|X) where X={a,b}, Y={x,y}
tensor1 = FXTensor(
    [[['a', 'b']], [['x', 'y']]],
    data=np.array([
        [0.2, 0.8],  # a -> x, y
        [0.6, 0.4]   # b -> x, y
    ])
)

# P(Z|Y) where Y={x,y}, Z={p,q}
tensor2 = FXTensor(
    [[['x', 'y']], [['p', 'q']]],
    data=np.array([
        [0.3, 0.7],  # x -> p, q
        [0.9, 0.1]   # y -> p, q
    ])
)

# Composition: P(Z|X) = P(Y|X) ; P(Z|Y)
result = tensor1.composition(tensor2)
assert result.labels == ([['a', 'b']], [['p', 'q']])
assert np.allclose(result.data, [
    [0.2*0.3 + 0.8*0.9, 0.2*0.7 + 0.8*0.1],  # a -> p, q
    [0.6*0.3 + 0.4*0.9, 0.6*0.7 + 0.4*0.1],  # b -> p, q
])
```

### Labeled Tensor Product

```python
# P(X) where X={a,b}
tensor1 = FXTensor(
    [[], [['a', 'b']]],
    data=np.array([0.3, 0.7])
)

# P(Y) where Y={x,y,z}
tensor2 = FXTensor(
    [[], [['x', 'y', 'z']]],
    data=np.array([0.2, 0.3, 0.5])
)

# Tensor product: P(X,Y) = P(X) ⊗ P(Y)
result = tensor1.tensor_product(tensor2)
assert result.labels == (None, [['a', 'b'], ['x', 'y', 'z']])
assert np.allclose(result.data, np.outer(
    np.array([0.3, 0.7]),
    np.array([0.2, 0.3, 0.5]),
))
```

## Simple Example: Weather Forecast (Labeled)

Model a weather system with states “Sunny” or “Rainy.”

- **State Tensor**: Represents today’s weather probability with labels. If today is certainly sunny, the state is `[1, 0]`.

  ```python
  weather_states = ['Sunny', 'Rainy']
  sunny_today = FXTensor([[], [weather_states]], data=np.array([1, 0]))
  ```

- **Process Tensor**: Represents a weather forecast as a labeled Markov kernel.

  ```python
  forecast_matrix = np.array([
      [0.8, 0.2],  # Sunny -> Sunny: 0.8, Rainy: 0.2
      [0.4, 0.6]   # Rainy -> Sunny: 0.4, Rainy: 0.6
  ])
  forecast_tensor = FXTensor([[weather_states], [weather_states]], data=forecast_matrix)
  ```

- **Composition**: Predict tomorrow’s weather by composing today’s state with the forecast.

  ```python
  sunny_tomorrow = sunny_today.composition(forecast_tensor)
  sunny_idx = sunny_tomorrow.get_label_index(0, 'Sunny')
  p_sunny = sunny_tomorrow.data[sunny_idx]  # 0.8
  ```

## Advanced Example: Multidimensional System (Labeled)

Model **Season** (Spring, Summer, Other) and **Weather** (Sunny, Rainy) given **Location** (Urban, Rural).

`conditionalization` and `jointification` require a **state** (empty domain). `marginalization` also works on a kernel.

### Kernel: P(Season, Weather | Location)

Profile `[[['Urban', 'Rural']], [['Spring', 'Summer', 'Other'], ['Sunny', 'Rainy']]]`, data shape `(2, 3, 2)`. Each location’s block sums to 1.

```python
location_labels = ['Urban', 'Rural']
season_labels = ['Spring', 'Summer', 'Other']
weather_labels = ['Sunny', 'Rainy']
process_data = np.array([
    [[0.2, 0.1], [0.3, 0.1], [0.2, 0.1]],  # Urban
    [[0.1, 0.2], [0.2, 0.2], [0.1, 0.2]],  # Rural
])
process_tensor = FXTensor(
    [[location_labels], [season_labels, weather_labels]],
    data=process_data,
)

# P(Season | Location) by summing out Weather
season_tensor = process_tensor.marginalization(start_B=2)
assert season_tensor.labels == ([['Urban', 'Rural']], [['Spring', 'Summer', 'Other']])
assert np.allclose(season_tensor.data, [
    [0.3, 0.4, 0.3],
    [0.3, 0.4, 0.3],
])
```

### Joint state: P(Location, Season, Weather)

A state has an empty domain. Split the last axis with `conditionalization(3)` to get `P(Weather | Location, Season)`.

```python
joint_data = np.array([
    [[0.08, 0.04], [0.12, 0.04], [0.08, 0.04]],  # Urban, total 0.4
    [[0.06, 0.12], [0.12, 0.12], [0.06, 0.12]],  # Rural, total 0.6
])
joint = FXTensor(
    [[], [location_labels, season_labels, weather_labels]],
    data=joint_data,
)

cond_tensor = joint.conditionalization(concat_start_index=3)
assert cond_tensor.labels == (
    [['Urban', 'Rural'], ['Spring', 'Summer', 'Other']],
    [['Sunny', 'Rainy']],
)
assert cond_tensor.is_markov()
assert np.allclose(cond_tensor.data, [
    [[2/3, 1/3], [0.75, 0.25], [2/3, 1/3]],
    [[1/3, 2/3], [0.50, 0.50], [1/3, 2/3]],
])
```

### Jointification of two states

Both arguments must be states. Empty-domain labels appear as `None`.

```python
location_state = FXTensor([[], [location_labels]], data=np.array([0.6, 0.4]))
traffic_labels = ['Low', 'High']
traffic_state = FXTensor([[], [traffic_labels]], data=np.array([0.7, 0.3]))
joint_state = location_state.jointification(traffic_state)
assert joint_state.labels == (None, [['Urban', 'Rural'], ['Low', 'High']])
assert np.allclose(joint_state.data, [
    [0.42, 0.18],
    [0.28, 0.12],
])
```

### Key Method Applications

#### `from_json(json_data)`

Creates a tensor from JSON data, loading profile and data to instantiate an FXTensor.

```python
json_data = {
    "profile": [[['a', 'b']], [['x', 'y']]],
    "data": [[0.2, 0.8], [0.6, 0.4]]
}
tensor = FXTensor.from_json(json_data)
assert tensor.labels == ([['a', 'b']], [['x', 'y']])
```

#### `from_strands(profile, strands)`

Creates a tensor from strands (sparse string representations of non-zero elements).

```python
profile = [[['a', 'b']], [['x', 'y', 'z']]]
strands = {
    "[[['a']], [['x']]]": 0.1,
    "[[['a']], [['y']]]": 0.2,
    "[[['a']], [['z']]]": 0.7,
    "[[['b']], [['x']]]": 0.3,
    "[[['b']], [['y']]]": 0.3,
    "[[['b']], [['z']]]": 0.4
}
tensor = FXTensor.from_strands(profile, strands)
assert tensor.labels == ([['a', 'b']], [['x', 'y', 'z']])
```

#### `identity_tensor(list_x)`

Creates an identity tensor for the given dimensions, supporting labeled or numeric inputs.

```python
labels = [['a', 'b']]
id_tensor = FXTensor.identity_tensor(labels)
assert id_tensor.labels == ([['a', 'b']], [['a', 'b']])
```

#### `copy_tensor(list_x, n=2)`

Creates a diagonal copy morphism `X → X^{⊗ n}`. `n=0` is discard, `n=1` is identity.

```python
copy = FXTensor.copy_tensor([2])
assert copy.profile == [[2], [2, 2]]
assert np.array_equal(copy.data, [
    [[1, 0],
     [0, 0]],
    [[0, 0],
     [0, 1]],
])

labeled_copy = FXTensor.copy_tensor([['a', 'b']])
assert labeled_copy.labels == ([['a', 'b']], [['a', 'b'], ['a', 'b']])
```

#### `unit_tensor(dims)`

Creates a unit state tensor (all-ones vector) for the given dimensions.

```python
dims = [2, 3]
unit = FXTensor.unit_tensor(dims)
assert unit.profile == [[], dims]
assert np.all(unit.data == 1)
```

#### `delta_tensor(dims)`

Creates a delta tensor (identity matrix) for the given dimensions, used for copying.

```python
dims = [2]
delta = FXTensor.delta_tensor(dims)
assert delta.profile == [[dims], [dims]]
```

## DisCoPy integration

`fxtensor-salmon` can convert labeled tensors to [DisCoPy](https://discopy.org) `Tensor` / `Box` values and interpret `discopy.markov` diagrams (Copy, Discard, Swap, Id) as `FXTensor` morphisms. This extra requires Python 3.10+ and is installed with:

```shell
pip install fxtensor-salmon[discopy]
```

Each profile factor is one wire. Unlabeled tensors map to `Dim`; labels are stored on Markov type names and restored when converting a `Box` back.

```python
from discopy.markov import Copy, Discard, Id, Swap, Ty
from fxtensor_salmon import FXTensor
from fxtensor_salmon.discopy import FXTensorFunctor, from_tensor, to_box, to_tensor

kernel = FXTensor(
    [[2], [2]],
    data=np.array([[0.8, 0.2], [0.4, 0.6]]),
)
assert from_tensor(to_tensor(kernel)) == kernel

weather = ['Sunny', 'Rainy']
forecast = FXTensor([[weather], [weather]], data=np.array([[0.8, 0.2], [0.4, 0.6]]))
box = to_box(forecast, name="forecast")

x, y = Ty('x'), Ty('y')
F = FXTensorFunctor({x: 2, y: 3}, {})
assert F(Copy(x)) == FXTensor.copy_tensor([2])
assert F(Discard(x)) == FXTensor.exclamation([2])
assert F(Swap(x, y)) == FXTensor.swap([2], [3])
assert F(Id(x)) == FXTensor.identity_tensor([2])
```

## Theoretical Background: Relation to Markov Categories

The `fxtensor-salmon` library is designed based on the **Markov Category**, a framework for categorical probability theory.

### Markov Category Basics

- **Objects**: State spaces, represented in `FXTensor` as `domain` or `codomain` (e.g., `[['Urban', 'Rural']]` or `[[2]]`).
- **Morphisms**: Markov kernels (probabilistic transitions), represented by `FXTensor` instances with profile and data.

### String diagrams

Diagrams are read top to bottom (inputs at the top, outputs at the bottom). The drawing conventions are referenced from T. Fritz (arXiv:1908.07021), implemented independently in TikZ. Boxes are morphisms, triangles are states `I → X`, a black dot branching downwards is `copy`, and a wire ending in a black dot is discard `!`. Sources are in [`docs/diagrams/`](https://github.com/salmonsnare/fxtensor-salmon/tree/v1.2.2/docs/diagrams/).

Copy is coassociative, counital and cocommutative:

<img src="https://raw.githubusercontent.com/salmonsnare/fxtensor-salmon/v1.2.2/docs/diagrams/copy_associativity.svg" alt="coassociativity">

<img src="https://raw.githubusercontent.com/salmonsnare/fxtensor-salmon/v1.2.2/docs/diagrams/copy_counitality.svg" alt="counitality">

<img src="https://raw.githubusercontent.com/salmonsnare/fxtensor-salmon/v1.2.2/docs/diagrams/copy_commutativity.svg" alt="cocommutativity">

Every morphism is Markov (discard is natural), `f ; ! = !` — in `FXTensor`, each row of a kernel sums to 1:

<img src="https://raw.githubusercontent.com/salmonsnare/fxtensor-salmon/v1.2.2/docs/diagrams/discard_naturality.svg" alt="f ; ! = !">

### Markov Category Operations

| Method | Role | When to use |
|---|---|---|
| `composition` | Sequential composition `A→B` then `B→C` | Wire processes in series, or apply a kernel to a state |
| `tensor_product` | Monoidal product | Place independent systems side by side |
| `marginalization` | Discard a suffix of the outputs | Sum out axes you no longer need |
| `conditionalization` | Joint state → kernel | Split a joint distribution into a conditional (states only) |
| `partial_composition` | Compose only some output wires | Keep a prefix of the outputs and feed the suffix into another kernel |
| `jointification` | Joint of two states | Combine two independent states (states only) |
| `conditional` | Kernel `A → X ⊗ Y` → `A ⊗ X → Y` | Conditionalize a kernel (generalizes `conditionalization`) |
| `bayesian_inversion` | `f: X → Y` and prior `I → X` → `f†: Y → X` | Bayesian update / posterior kernel |
| `almost_surely_equal` | `f = g` p-a.s. | Compare kernels only where the prior has mass |
| `is_absolutely_continuous` | `p ≪ q` | Check support inclusion of states / kernels |
| `exclamation` | Discard morphism `X → I` | Build an all-ones discarding tensor |
| `copy_tensor` | Copy morphism `X → X^{⊗ n}` | Deterministic copy / diagonal (`n=0` is discard) |
| `delta_tensor` | Identity (alias of `identity_tensor`) | Build an identity kernel |

#### `composition` — sequential wiring

Use when the codomain of `f` matches the domain of `g`. Same numbers as the labeled example above: `P(Z|X) = P(Y|X) ; P(Z|Y)`.

```python
result = tensor1.composition(tensor2)
assert np.allclose(result.data, [[0.78, 0.22], [0.54, 0.46]])
```

<img src="https://raw.githubusercontent.com/salmonsnare/fxtensor-salmon/v1.2.2/docs/diagrams/composition.svg" alt="f ; g">

#### `tensor_product` — independent systems in parallel

<img src="https://raw.githubusercontent.com/salmonsnare/fxtensor-salmon/v1.2.2/docs/diagrams/tensor_product.svg" alt="f ⊗ g">

Use to put two morphisms (or two states) next to each other without coupling.

```python
px = FXTensor([[], [['a', 'b']]], data=np.array([0.3, 0.7]))
py = FXTensor([[], [['x', 'y', 'z']]], data=np.array([0.2, 0.3, 0.5]))
pxy = px.tensor_product(py)
assert pxy.labels == (None, [['a', 'b'], ['x', 'y', 'z']])
assert np.allclose(pxy.data, [[0.06, 0.09, 0.15], [0.14, 0.21, 0.35]])
```

#### `marginalization` — drop a suffix of the outputs

Use `start_B` (1-based) as the first codomain axis to sum out. A state with profile `[[], [2, 3]]` and `start_B=2` keeps the first axis.

```python
state = FXTensor([[], [2, 3]], data=np.array([
    [0.1, 0.2, 0.3],
    [0.15, 0.05, 0.2],
]))
marginal = state.marginalization(2)
assert marginal.profile == [[], [2]]
assert np.allclose(marginal.data, [0.6, 0.4])
```

#### `conditionalization` — joint state to a kernel

Use only on a state. `concat_start_index` (1-based) is the first axis of the new codomain. A zero slice stays zero.

```python
joint_state = FXTensor([[], [2, 2]], data=np.array([[0.1, 0.2], [0.0, 0.0]]))
kernel = joint_state.conditionalization(2)
assert kernel.profile == [[2], [2]]
assert kernel.is_markov()
assert np.allclose(kernel.data, [[1/3, 2/3], [0.0, 0.0]])
```

#### `partial_composition` — compose only the trailing outputs

For `f: A → B ⊗ C` and `g: C → D`, `f.partial_composition(g, 2)` keeps `B` and composes on `C`, giving `A → B ⊗ D`.

```python
f = FXTensor([[2], [2, 2]], data=np.array([
    [[1.0, 0.0], [0.0, 1.0]],
    [[0.0, 1.0], [1.0, 0.0]],
]))
g = FXTensor([[2], [2]], data=np.array([
    [0.2, 0.8],
    [0.6, 0.4],
]))
partial = f.partial_composition(g, 2)
assert partial.profile == [[2], [2, 2]]
assert np.allclose(partial.data, [
    [[0.2, 0.8], [0.6, 0.4]],
    [[0.6, 0.4], [0.2, 0.8]],
])
```

`concat_start_index=1` composes on the whole codomain (the case covered by the tests).

#### `jointification` — two states into one joint

Same numbers as the tensor product of states; both domains must be empty.

```python
px = FXTensor([[], [['a', 'b']]], data=np.array([0.3, 0.7]))
py = FXTensor([[], [['x', 'y', 'z']]], data=np.array([0.2, 0.3, 0.5]))
joint_xy = px.jointification(py)
assert joint_xy.labels == (None, [['a', 'b'], ['x', 'y', 'z']])
assert np.allclose(joint_xy.data, [[0.06, 0.09, 0.15], [0.14, 0.21, 0.35]])
```

#### `bayesian_inversion` — posterior kernel

<img src="https://raw.githubusercontent.com/salmonsnare/fxtensor-salmon/v1.2.2/docs/diagrams/bayesian_inversion.svg" alt="p;copy;(id⊗f) = p;f;copy;(f†⊗id)">

For a prior `p: I → X` and a kernel `f: X → Y`, `f.bayesian_inversion(p)` returns `f†: Y → X` with `f†(x|y) = p(x) f(y|x) / Σ p(x') f(y|x')` (Cho & Jacobs). It satisfies `p ; copy ; (id ⊗ f) = p ; f ; copy ; (f† ⊗ id)`. Outputs with zero evidence stay zero.

```python
health = ['Sick', 'Healthy']
result = ['Positive', 'Negative']
prior = FXTensor([[], [health]], data=np.array([0.01, 0.99]))
test = FXTensor([[health], [result]], data=np.array([[0.9, 0.1], [0.05, 0.95]]))
posterior = test.bayesian_inversion(prior)
assert posterior.labels == ([result], [health])
assert np.isclose(posterior.data[0, 0], 2 / 13)  # P(Sick | Positive)
```

#### `conditional` — conditionalize a kernel

`f.conditional(k)` turns `f: A → X ⊗ Y` into `A ⊗ X → Y`, where `k` (1-based) is the first codomain axis of `Y`. On a state it equals `conditionalization(k)`.

```python
f = FXTensor([[['a']], [['x0', 'x1'], ['y0', 'y1']]], data=np.array([[[0.1, 0.3], [0.2, 0.4]]]))
c = f.conditional(2)
assert c.labels == ([['a'], ['x0', 'x1']], [['y0', 'y1']])
assert np.allclose(c.data, [[[0.25, 0.75], [1/3, 2/3]]])
```

#### `support`, `almost_surely_equal`, `is_absolutely_continuous` — a.s. reasoning

<img src="https://raw.githubusercontent.com/salmonsnare/fxtensor-salmon/v1.2.2/docs/diagrams/almost_surely_equal.svg" alt="p;copy;(id⊗f) = p;copy;(id⊗g)">

`support()` returns the 0/1 indicator of nonzero entries (same profile and labels). `f.almost_surely_equal(g, p)` checks Fritz's `p ; copy ; (id ⊗ f) = p ; copy ; (id ⊗ g)`, i.e. `f` and `g` agree on the support of `p`. `p.is_absolutely_continuous(q)` checks `p ≪ q` (zero in `q` implies zero in `p`; input-wise for kernels, cf. arXiv:2308.00651). Zero is judged with `_ATOL`; labels are not compared.

```python
prior = FXTensor([[], [['x0', 'x1', 'x2']]], data=np.array([0.4, 0.6, 0.0]))
f = FXTensor([[['x0', 'x1', 'x2']], [['y0', 'y1']]], data=np.array([[0.2, 0.8], [0.5, 0.5], [1.0, 0.0]]))
g = FXTensor([[['x0', 'x1', 'x2']], [['y0', 'y1']]], data=np.array([[0.2, 0.8], [0.5, 0.5], [0.3, 0.7]]))
assert np.array_equal(prior.support().data, [1.0, 1.0, 0.0])
assert f.almost_surely_equal(g, prior)   # f and g differ only on the null point x2
assert f.is_absolutely_continuous(g)     # f(·|x) ≪ g(·|x) for every x
assert not g.is_absolutely_continuous(f)
```

### Probabilistic Properties

- `is_markov()`: Verifies if the tensor satisfies the normalization condition (sum of outputs equals 1 or 0). States (empty domain) always return `False`; check `np.isclose(state.data.sum(), 1)` for a normalized state.
- `is_deterministic()`: True if every codomain slice is a point mass (one-hot), i.e. `f ; copy = copy ; (f ⊗ f)`. Zero slices are not deterministic.
- Labeled tensors enable intuitive interpretation via `get_label_index` and `get_index_label`.

## Testing

Tests are implemented in `tests/test_fxtensor.py` using `pytest`.

```bash
pytest
```

## References
- [1] [檜山正幸のキマイラ飼育記 (はてなBlog), マルコフ圏 A First Look -- 圏論的確率論の最良の定式化](https://m-hiyama.hatenablog.com/entry/2020/06/09/154044)
- [2] [檜山正幸のキマイラ飼育記 (はてなBlog), マルコフ圏におけるテンソル計算の手順とコツ](https://m-hiyama.hatenablog.com/entry/2021/04/05/153325)
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