---
kind: concept
slug: monotonic-alignment-search
title: Monotonic alignment search
summary: >-
  Finds a monotonic, surjective alignment between text tokens and spectrogram frames by maximising the frames' likelihood under each token's predicted mean, so no external aligner is needed. Known from one paper card, Grad-TTS; its search procedure is not described there.
tags: [alignment, text-to-speech]
papers: ["2105.06337"]
related: [dynamic-time-warping, diffusion-models]
status: draft
updated_at: 2026-10-18
---

## Overview

Text-to-speech models must know how many output frames each input token
covers. Monotonic alignment search (MAS) is a way to learn that alignment
during training. Given the encoder's per-token means and the target
mel-spectrogram, it returns the monotonic, surjective mapping from frames to
tokens that makes the mel frames most likely under Gaussians centred on those
means [arxiv:2105.06337 §3.2] [arxiv:2105.06337 eq. 14]. The frame counts per
token then serve as duration targets for a duration predictor
[arxiv:2105.06337 §3.2].

The only source card here is Grad-TTS, which uses MAS in this way. The card
attributes the use of MAS for this purpose to Glow-TTS (Kim et al., NeurIPS
2020) [arxiv:2105.06337 §1]. That paper has no card on this wiki, so it is
named but not described.

## Variants

Only one variant is documented here.

### MAS with a Gaussian likelihood, as in Grad-TTS

The objective is the log-likelihood of each mel frame under a unit-variance
Gaussian centred on the mean of the token assigned to that frame. The search
keeps the alignment that maximises the total. Training alternates between
searching the alignment and updating the parameters. The encoder loss is
included, and the authors report that without it the model did not learn an
alignment [arxiv:2105.06337 §3.2]. Card: [Grad-TTS: A Diffusion Probabilistic Model for Text-to-Speech](/papers/2105.06337).

## When to use which

The source card supports only one comparison: a model that predicts durations
without an alignment search needs durations from a pre-computed teacher
alignment, whereas MAS learns the alignment jointly with the model
[arxiv:2105.06337 §1]. Beyond that, no constraint-based guidance can be drawn
from the cards on this wiki.

| Situation | Use | Why |
|---|---|---|
| No external aligner and no pre-computed durations | MAS, as in Grad-TTS | Alignment is learned during training [arxiv:2105.06337 §1] [arxiv:2105.06337 §3.2] |
| An external aligner or teacher durations already exist | Not stated | No card compares MAS with a pre-computed teacher alignment on results |
| Need for a different alignment objective or a faster search | Not stated | The card does not compare alternative search methods |

## Implementation sketch

The card does not describe the search procedure itself, so the function
`monotonic_align_search` below is a placeholder. Only its inputs, outputs and
the way the alignment is used are taken from the source.

```pseudocode
def train_step_alignment(token_means, mels, mel_lengths):
    # Likelihood of each frame under each token's mean, unit variance  [§3.2]
    # token_means: (tokens, 80); mels: (80, frames)
    # Search the monotonic, surjective alignment with the highest total
    # log-likelihood  [§3.2] [eq. 14]
    alignment = monotonic_align_search(token_means, mels, mel_lengths)
    # Frames per token: duration targets for the duration predictor  [§3.2]
    frames_per_token = count_frames_per_token(alignment)
    return alignment, frames_per_token
```

Points where variants differ, none of which are established by a second
card: the normalisation of the alignment term in the loss, and whether the
encoder loss is included. See Contradictions.

## Contradictions

The Grad-TTS card records a difference between the paper and the released
code for the encoder loss that the alignment search uses. The paper's
equation 14 is an unnormalised sum over frames and mel bins [arxiv:2105.06337 eq. 14].
The released code averages the same expression over valid frames and bins,
which scales the term down relative to the paper's equation
[code: Grad-TTS/model/tts.py#L178-L179]. The two readings weight the alignment
term differently against the other losses. The card does not resolve which
is intended.

## Papers

- [Grad-TTS: A Diffusion Probabilistic Model for Text-to-Speech](/papers/2105.06337): uses MAS to learn alignments for a diffusion-based text-to-speech model and sets the encoder loss that the alignment depends on [arxiv:2105.06337 §3.2].