Input-Dependent Transition Matrices

• Input-dependent transition matrices are defined as matrix-valued functions whose entries vary with external inputs, enabling context-aware and dynamic modeling.
• They can be constructed using linear parameterizations, deep networks, or mixture representations, which balance adaptability with tractable inference.
• Their application spans sequential neural networks, label noise modeling, and ecological dynamics, where adaptive state transitions are crucial to performance.

Input-dependent transition matrices are matrix-valued functions whose entries vary with respect to external inputs, observed features, or side information. This paradigm is central in modern probabilistic modeling, sequential learning architectures, dynamical systems, and noise modeling, reflecting the need for transition dynamics that adapt to contextual or environmental variables.

1. Formal Definition and General Properties

A transition matrix $P(x)$ in a discrete Markov process is input-dependent if its entries $p_{ij}(x)$ satisfy

$$p_{ij}(x) = \mathbb{P}(X_{t+1} = j \mid X_t = i, \text{Input } x_t = x),$$

where $x$ may be a covariate, observation, time-varying parameter, or instance feature. For continuous-time Markov chains, the rate matrix $Q(x)$ expresses $q_{ij}(x)$ as a function of input.

Input-dependent matrices generalize the homogeneous case ($P$ fixed for all $x$) by allowing $P$ or $Q$ to vary, often in high-dimensional or nonparametric ways. Their semantics depend heavily on domain-specific modeling assumptions: in sequential neural architectures, $A(x_t)$ encodes both adaptation to symbol sequence and recurrence dynamics (Khavari et al., 10 Aug 2025), while in label noise modeling, $T(x)$ captures annotation noise conditional on data features (Xia et al., 2020, Li et al., 2023).

2. Methods of Construction and Parameterization

(a) Function-based Parameterizations

Transition matrices may be specified as explicit functions of input, such as:

• Linear/affine: $P(x) = P_0 + x P_1$
• Deep networks: $P(x) = \text{NN}_\theta(x)$ (used in instance-dependent label noise (Li et al., 2023))
• Multi-task Gaussian Process regression: $a_{ij}(t) = f_{ij}(z(t-1), \beta_{ij})$ for time-varying transitions (Ugurel, 2023)

(b) Mixture Representations

A widely used approach decomposes $P(x)$ as a convex combination:

$P(x) = \sum_{j=1}^{r} h_j(x) P^j$

where $P^j$ are part-dependent transition matrices and $h_j(x)$ are non-negative weights summing to one, typically learned via Non-negative Matrix Factorization (NMF) on the input feature space (Xia et al., 2020). This facilitates tractable approximation of the complex instance-dependent transition matrix $T(x)$ and is robust in high-noise regimes.

(c) Hierarchical and Bayesian Models

Imprecise probabilistic approaches use prior parameter sets $(s, A(x))$ in an estimator such as

$q_{xy}(x) = \frac{s(x) A(x, y) + n_{xy}}{d_x}$

with diagonal terms set for row sum constraints (Krak et al., 2018). Prior parameters may themselves be functions of input or side information.

3. Mathematical Implications and Key Theorems

Input-dependent transition matrices lead to nonstationary or context-aware dynamics. Some notable results:

• In Markov chains, stationary distributions and mean first passage times become explicit functions of input through the column sums $c_j(x)$ and the corresponding generalized inverse $H(x)$ (Hunter, 2011):

$\pi_j(x) = \sum_i c_i(x) h_{ij}(x)$

Changing column sums (i.e., varying transition probabilities with input) directly alters stationary behaviors and system properties such as Kemeny’s constant.

• For LRNN models, solving the parity or modular state-tracking tasks requires a single recurrence layer whose transition matrix is both input-dependent and possesses negative or complex eigenvalues (Khavari et al., 10 Aug 2025). A purely input-independent or non-negative SSM is provably insufficient.
• In random walk and graph models, the transition matrices constructed from adjacency matrices inherit input dependence through the graph structure and base-point selection (Ikkai et al., 2020).
• In continuous-time Markov embedding, a unique intensity matrix $Q(x)$ can be constructed for each input-dependent transition matrix $P(x)$ via conditional embedding, using fixed-point equations parameterized by input (Carette et al., 2023).

4. Practical Applications and Design Considerations

Input-dependent transition matrices are exploited in various fields:

Application Area	Role of Input-dependent Matrix	Notable References
Sequential neural networks, SSMs	Modulate state evolution by input	(Khavari et al., 10 Aug 2025)
Instance/annotator-dependent label noise	Model context-specific noise	(Xia et al., 2020, Li et al., 2023)
Population/ecological modeling	Capture migration rates/flows by covariates	(Goswami, 2022)
Constrained Markov process (mobility)	Learn time-/location-varying transitions	(Ugurel, 2023)
Credit risk/migration modeling	Adapt generator to economic inputs	(Carette et al., 2023)

Key considerations when designing input-dependent transition matrices:

• Constraint enforcement: Stochasticity and non-negativity must be maintained for all inputs; constrained optimization or pointwise control is used in GPs (Ugurel, 2023).
• Parameter sharing: In high-dimensional cases (e.g., annotator-instance noise), knowledge transfer and neighbor calibration via deep networks and GCNs mitigate overfitting and annotation sparsity (Li et al., 2023).
• Robust inference: Bayesian and imprecise probabilistic estimators adapt prior parameters to input, allowing for set-valued and robust parameter updates, especially in data-scarce regimes (Krak et al., 2018).
• Spectral analysis: For population dynamics, the eigenstructure of the transition matrix as determined by input informs long-term stable states and transient patch identification (Goswami, 2022); irreducible/reducible decomposition can reflect population viability.

5. Comparative Analysis and Limitations

Compared to input-independent matrices, input-dependent transition matrices offer:

• Greater modeling fidelity for complex, context-specific systems.
• Ability to capture state-tracking, alternation, and dynamic adaptation in tasks (e.g., parity, modular counting, context-aware prediction).
• Enhanced robustness to domain-specific noise, particularly in annotation and crowdsourcing contexts.

However, increased flexibility introduces estimation and computational challenges:

• Larger parameter space (for instance- and annotator-dependence, O(r × n × C × C) for TAIDTM (Li et al., 2023)) raises risk of overfitting.
• Constraint satisfaction (non-negativity, stochasticity, spectral properties) is computationally more demanding, especially when implemented via kernel methods or deep architectures (Ugurel, 2023).
• For some tasks, distributing input dependence and negative eigenvalues across separate layers is insufficient; a single unified recurrence layer must possess both properties (Khavari et al., 10 Aug 2025).

6. Representative Mathematical Formulations

Notable formulations illustrating input dependence include:

• SSM recurrence with input-dependent transition:

$h_t = A(x_t) h_{t-1} + B(x_t) x_t$

• Instance-dependent noise transition approximation via parts:

$T(x) \approx \sum_{j=1}^r h_j(x) P^j$

• Imprecise CTMC estimator:

$q_{xy}(x) = \frac{s(x) A(x, y) + n_{xy}}{d_x}$

• Time-variability (mobility modeling):

$a_{ij}(t) = f_{ij}(z(t-1), \beta_{ij})$

These constructions require enforcement of matrix constraints and often exploit parametric, nonparametric, or mixture-based approaches to balance flexibility, tractability, and interpretability.

7. Impact and Future Directions

The shift toward input-dependent transition matrices reflects the increasing recognition that system dynamics, noise processes, and sequential behavior in complex domains are rarely homogeneous. Theoretical results confirm that input dependence and spectral diversity (including negative/complex eigenvalues) are indispensable for full expressive power in stateful models (Khavari et al., 10 Aug 2025). Empirical advances in deep learning, kernel methods, and Bayesian estimation have enabled scalable and robust algorithmic frameworks for high-dimensional input-dependent transition matrices (Ugurel, 2023, Li et al., 2023). Further research will likely focus on efficient parameter sharing, scalable constraint enforcement, and the integration of these techniques into broader frameworks for sequential decision making, probabilistic inference, and robust learning from noisy real-world data.