YouTip

## PyTorch torch.arccosh `torch.arccosh` is a PyTorch function used to compute the element-wise inverse hyperbolic cosine (also known as $\text{arcosh}$ or $\cosh^{-1}$) of an input tensor. This function is an alias for `torch.acosh`. Both functions yield identical results and can be used interchangeably. --- ### Mathematical Definition For a given input $x$, the inverse hyperbolic cosine is defined mathematically as: $$\text{arccosh}(x) = \ln\left(x + \sqrt{x^2 - 1}\right)$$ * **Domain:** $[1, \infty)$ * **Codomain:** $[0, \infty)$ > **Note:** Since the domain of the inverse hyperbolic cosine function is restricted to real numbers greater than or equal to 1, passing values less than 1 will result in `NaN` (Not a Number) for real-valued tensors, or complex numbers if the input tensor is complex. --- ### Syntax ```python torch.arccosh(input, *, out=None) -> Tensor ``` #### Parameters | Parameter | Type | Description | | :--- | :--- | :--- | | `input` | *Tensor* | The input tensor containing values $\ge 1$. | | `out` | *Tensor (Optional)* | The output tensor. Defaults to `None`. | #### Returns * **Tensor**: A tensor of the same shape and data type as the `input` containing the element-wise inverse hyperbolic cosine values. --- ### Code Examples #### Example 1: Basic Usage with Valid Inputs (Real Numbers $\ge 1$) ```python import torch # Create a tensor with values greater than or equal to 1 x = torch.tensor([1.0, 1.5, 2.0, 3.0]) # Compute the inverse hyperbolic cosine result = torch.arccosh(x) print("Input Tensor:") print(x) print("\nResult Tensor:") print(result) ``` **Output:** ```text Input Tensor: tensor([1.0000, 1.5000, 2.0000, 3.0000]) Result Tensor: tensor([0.0000, 0.9624, 1.3170, 1.7627]) ``` --- #### Example 2: Handling Out-of-Domain Inputs (Values $< 1$) If you pass values strictly less than 1, PyTorch will return `nan` for those elements. ```python import torch # Create a tensor containing values less than 1 x = torch.tensor([0.5, 1.0, 2.0]) # Compute arccosh result = torch.arccosh(x) print(result) ``` **Output:** ```text tensor([ nan, 0.0000, 1.3170]) ``` --- #### Example 3: In-Place Operation If you want to perform the operation in-place to save memory, you can use `torch.arccosh_` (with a trailing underscore) or use the `out` parameter. ```python import torch x = torch.tensor([1.0, 2.0, 3.0]) # In-place modification x.arccosh_() print(x) ``` **Output:** ```text tensor([0.0000, 1.3170, 1.7627]) ``` --- ### Considerations & Best Practices 1. **Alias Equivalence**: `torch.arccosh` and `torch.acosh` are completely identical in behavior and performance. You can use either depending on your naming preference. 2. **Domain Constraints**: Always ensure your input values are $\ge 1.0$ when working with real-valued tensors to avoid unexpected `nan` values in your pipeline. If your model's forward pass can produce values $< 1$, consider using `torch.clamp(input, min=1.0)` before applying `torch.arccosh`. 3. **Complex Number Support**: If the input tensor is complex, `torch.arccosh` will compute the complex inverse hyperbolic cosine, which extends the domain to all complex numbers.