Inverse Blahut-Arimoto Algorithm (IBAA)

• Inverse Blahut-Arimoto Algorithm (IBAA) is a reverse framework that inverts standard rate-distortion updates to recover unknown distortion functions from observed conditional strategies.
• It encompasses backward recursions and reverse projections, including causal and information-geometric formulations, to optimize directed information and channel capacity.
• IBAA has practical applications in biological decision models, chemotaxis, and various channel coding scenarios, illustrating its versatility in information-theoretic problems.

Searching arXiv for recent and foundational papers on the Inverse Blahut-Arimoto Algorithm and closely related Blahut-Arimoto inverse/backward variants. {"query":"all:(\"Inverse Blahut-Arimoto\" OR IBAA OR \"Backward em-algorithm\" OR \"Extension of the Blahut-Arimoto algorithm for maximizing directed information\" OR \"Geometry of Arimoto Algorithm\" OR \"Iterative minimization algorithm on a mixture family\" OR \"Bregman-divergence-based Arimoto-Blahut algorithm\" OR \"Petz-Rényi capacity\" )","max_results":10,"sort_by":"submittedDate","sort_order":"descending"} I found several relevant arXiv papers, including explicit IBAA usage and broader backward/generalized Blahut-Arimoto formulations:

• "Inferring the Chemotaxis Distortion Function from Cellular Decision Strategies" (Vakilipoor et al., 30 Oct 2025)
• "Extended AB Algorithms for Bistatic Integrated Sensing and Communications Systems" (Jiao et al., 11 Aug 2025)
• "A Mirror-Descent Algorithm for Computing the Petz-Rényi Capacity of Classical-Quantum Channels" (Lai et al., 15 Jan 2026)
• "Bregman-divergence-based Arimoto-Blahut algorithm" (Hayashi, 2024)
• "Probabilistic Shaping for the AWGN Channel" (Delsad, 2023)
• "Iterative minimization algorithm on a mixture family" (Hayashi, 2023)
• "Computing the Classical-Quantum channel capacity..." (Li et al., 2019)
• "Geometry of Arimoto Algorithm" (Toyota, 2019)
• "Blahut-Arimoto Algorithm and Code Design for Action-Dependent Source Coding Problems" (Trillingsgaard et al., 2013)
• "Extension of the Blahut-Arimoto algorithm for maximizing directed information" (Naiss et al., 2010) Inverse Blahut-Arimoto Algorithm (IBAA) denotes inverse, backward, or target-constrained counterparts of the Blahut-Arimoto (BA) framework. In its most explicit form, IBAA inverts the standard rate-distortion BA update to recover a distortion function $d(x,y)$ from an observed conditional strategy $P_{Y|X}(y|x)$ and output marginal $P_Y(y)$ (Vakilipoor et al., 30 Oct 2025). In adjacent literatures, closely related constructions appear as a backward-index alternating-maximization algorithm for directed information (Naiss et al., 2010), an information-geometric Backward em-algorithm whose approximation yields the Arimoto recursion (Toyota, 2019), and generalized BA-type projection or mirror-descent methods on mixture families, Bregman systems, and classical-quantum optimization domains (Hayashi, 2023, Hayashi, 2024, Lai et al., 15 Jan 2026). Taken together, these works indicate that IBAA is best understood as a family of inverse BA viewpoints rather than a single canonical iteration.

1. Classical BA and the inverse viewpoint

The classical BA paradigm solves a forward optimization problem: either maximize mutual information for channel capacity or minimize mutual information subject to a distortion constraint in rate-distortion theory. In the rate-distortion setting used by the explicit IBAA paper, the forward problem is

$$P_{Y|X}^{\star}(y|x)=\operatorname*{arg\,min}_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y),$$

with rate-distortion function

$$R(D)=\min_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y).$$

The associated BA update is

$$P_{Y|X}(y|x)=\frac{P_Y(y)e^{-\lambda d(x,y)}}{\sum_{y'}P_Y(y')e^{-\lambda d(x,y')}}.$$

The inverse problem reverses this direction: instead of computing an optimal strategy from a known distortion or known channel parameters, it seeks to recover the latent criterion, multiplier, or structured distribution that makes an observed strategy optimal (Vakilipoor et al., 30 Oct 2025).

Two broader inverse interpretations recur in the literature. One treats BA inversely in time or index order, as in maximizing directed information over causally conditioned laws $p(x^n\|y^{n-1})$ by a backward recursion from time $n$ to $1$ (Naiss et al., 2010). Another treats BA inversely in an information-geometric sense: instead of projecting toward lower KL divergence, one constructs a backward or reverse procedure that increases KL divergence and thereby approaches channel capacity (Toyota, 2019). This broader usage explains why several papers are IBAA-relevant even when they do not use the term explicitly.

2. Explicit inversion of the rate-distortion update

The clearest closed-form IBAA is the one introduced for biological decision strategies. Its premise is that the conditional strategy $P_{Y|X}(y|x)$ and the marginal $P_{Y|X}(y|x)$0 are observed, whereas the distortion function optimized by the system is unknown. Starting from the BA fixed-point relation,

$P_{Y|X}(y|x)$1

rearrangement yields

$P_{Y|X}(y|x)$2

Setting $P_{Y|X}(y|x)$3 for simplicity gives the operational inverse estimate

$P_{Y|X}(y|x)$4

followed by the normalization

$P_{Y|X}(y|x)$5

which enforces nonnegativity (Vakilipoor et al., 30 Oct 2025).

The operational procedure is direct rather than variational. One first estimates the joint distribution with Laplace smoothing,

$P_{Y|X}(y|x)$6

then computes $P_{Y|X}(y|x)$7 and $P_{Y|X}(y|x)$8, forms $P_{Y|X}(y|x)$9, and shifts by the per-$P_Y(y)$0 minimum. The paper emphasizes that this is an inverse estimator, not an optimization over $P_Y(y)$1 (Vakilipoor et al., 30 Oct 2025).

The principal identifiability limitations are explicit. Because BA depends on $P_Y(y)$2, the recovered distortion is identifiable only up to a positive multiplicative constant unless $P_Y(y)$3 is known. It is also identifiable only up to an additive offset depending on $P_Y(y)$4, since $P_Y(y)$5 leaves the conditional strategy unchanged. Zero empirical counts would imply infinite distortions for feasible events, so Laplace smoothing is required (Vakilipoor et al., 30 Oct 2025).

Validation in the same paper proceeds in two settings. In a binary apoptosis model following Porter et al., IBAA reconstructs the shape of both a Hamming-like distortion and a rectified squared distortion from forward BAA-generated strategies. In a chemotaxis application based on the LEGI model, the inferred distortion is smallest when the movement direction $P_Y(y)$6 is close to the source direction $P_Y(y)$7, and the averaged distortion resembles the cosine-like distortion used in earlier optimal gradient-sensing work. The paper further reports that larger Hill coefficient $P_Y(y)$8 produces a distortion function with larger magnitude and steeper slope, which it interprets as a state-dependent decision criterion (Vakilipoor et al., 30 Oct 2025).

3. Backward-index and information-geometric inverse formulations

A distinct inverse BA meaning appears in the extension of BA to directed information. For a fixed causal channel law

$P_Y(y)$9

the objective is

$$P_{Y|X}^{\star}(y|x)=\operatorname*{arg\,min}_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y),$$0

The algorithm alternates between updating $$P_{Y|X}^{\star}(y|x)=\operatorname*{arg\,min}_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y),$$1 by a backward-index maximization, starting from $$P_{Y|X}^{\star}(y|x)=\operatorname*{arg\,min}_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y),$$2 and moving down to $$P_{Y|X}^{\star}(y|x)=\operatorname*{arg\,min}_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y),$$3, and updating

$$P_{Y|X}^{\star}(y|x)=\operatorname*{arg\,min}_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y),$$4

It also produces lower and upper bounds $$P_{Y|X}^{\star}(y|x)=\operatorname*{arg\,min}_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y),$$5 and $$P_{Y|X}^{\star}(y|x)=\operatorname*{arg\,min}_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y),$$6, with stopping rule $$P_{Y|X}^{\star}(y|x)=\operatorname*{arg\,min}_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y),$$7, and the bounds satisfy the sandwich behavior $$P_{Y|X}^{\star}(y|x)=\operatorname*{arg\,min}_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y),$$8, $$P_{Y|X}^{\star}(y|x)=\operatorname*{arg\,min}_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y),$$9 (Naiss et al., 2010). The paper explicitly characterizes this as “inverse” because the causal optimization must be solved by backward maximization over time rather than by a single forward maximization over $$R(D)=\min_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y).$$0. For $$R(D)=\min_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y).$$1, it reduces exactly to regular BAA (Naiss et al., 2010).

An information-geometric variant appears in the “Backward em-algorithm.” There the channel-capacity problem is rewritten on the joint-distribution manifold

$$R(D)=\min_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y).$$2

and capacity becomes

$$R(D)=\min_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y).$$3

The backward e-step seeks $$R(D)=\min_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y).$$4 whose $$R(D)=\min_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y).$$5-projection returns the current point in $$R(D)=\min_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y).$$6, and the exact condition is

$$R(D)=\min_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y).$$7

The backward m-step is then defined by a reverse projection condition, and the paper proves the monotonicity relation

$$R(D)=\min_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y).$$8

Because the backward m-step is generally hard to solve exactly, the paper introduces an approximation under which the resulting update is exactly the standard Arimoto recursion (Toyota, 2019). The term IBAA does not appear there, but the construction is a direct geometric precursor of reverse or inverse BA.

4. Target-distortion, dual, and constrained inverse BA constructions

A third IBAA-related usage arises when BA is inverted with respect to its Lagrange parametrization. In bistatic ISAC, the optimization seeks the maximum rate subject to a prescribed sensing distortion $$R(D)=\min_{P_{Y|X}(y|x):\,\mathbb{E}[d(X,Y)]\le D} I(X;Y).$$9, but the distortion constraints are non-convex in $$P_{Y|X}(y|x)=\frac{P_Y(y)e^{-\lambda d(x,y)}}{\sum_{y'}P_Y(y')e^{-\lambda d(x,y')}}.$$0. The extended AB algorithms introduce auxiliary variables $$P_{Y|X}(y|x)=\frac{P_Y(y)e^{-\lambda d(x,y)}}{\sum_{y'}P_Y(y')e^{-\lambda d(x,y')}}.$$1 for squared error and $$P_{Y|X}(y|x)=\frac{P_Y(y)e^{-\lambda d(x,y)}}{\sum_{y'}P_Y(y')e^{-\lambda d(x,y')}}.$$2 for log-loss, converting the non-convex constraints into linear constraints in $$P_{Y|X}(y|x)=\frac{P_Y(y)e^{-\lambda d(x,y)}}{\sum_{y'}P_Y(y')e^{-\lambda d(x,y')}}.$$3. The resulting updates have closed form, for example

$$P_{Y|X}(y|x)=\frac{P_Y(y)e^{-\lambda d(x,y)}}{\sum_{y'}P_Y(y')e^{-\lambda d(x,y')}}.$$4

in the squared-error case, with $$P_{Y|X}(y|x)=\frac{P_Y(y)e^{-\lambda d(x,y)}}{\sum_{y'}P_Y(y')e^{-\lambda d(x,y')}}.$$5 determined implicitly by the distortion-boundary equation $$P_{Y|X}(y|x)=\frac{P_Y(y)e^{-\lambda d(x,y)}}{\sum_{y'}P_Y(y')e^{-\lambda d(x,y')}}.$$6. The paper explicitly frames this as more “inverse-like” than sweeping a multiplier to trace an entire rate-distortion curve, because it solves for the maximizing distribution at a target distortion point (Jiao et al., 11 Aug 2025).

A closely related dual-parametric viewpoint appears in action-dependent source coding. Building on the characterization restated from Permuter et al., the paper introduces Lagrange multipliers $$P_{Y|X}(y|x)=\frac{P_Y(y)e^{-\lambda d(x,y)}}{\sum_{y'}P_Y(y')e^{-\lambda d(x,y')}}.$$7 and $$P_{Y|X}(y|x)=\frac{P_Y(y)e^{-\lambda d(x,y)}}{\sum_{y'}P_Y(y')e^{-\lambda d(x,y')}}.$$8 for distortion and action cost and writes

$$P_{Y|X}(y|x)=\frac{P_Y(y)e^{-\lambda d(x,y)}}{\sum_{y'}P_Y(y')e^{-\lambda d(x,y')}}.$$9

Its BA-type algorithm alternates over $p(x^n\|y^{n-1})$0, $p(x^n\|y^{n-1})$1, and $p(x^n\|y^{n-1})$2, while the optimal strategy distribution is recovered through a nested dual and fixed-point scheme. This is the paper’s key inverse or dual representation: boundary points $p(x^n\|y^{n-1})$3 are reconstructed by tuning $p(x^n\|y^{n-1})$4 and solving the BA-type minimization (Trillingsgaard et al., 2013).

In probabilistic shaping for the AWGN channel, standard BA optimizes the input distribution for a fixed constellation and fixed noise variance, whereas fixed-SNR design requires a constrained BA with gain $p(x^n\|y^{n-1})$5 and power constraint

$p(x^n\|y^{n-1})$6

The inner update depends on a Lagrange multiplier $p(x^n\|y^{n-1})$7 determined numerically from a scalar root condition, and an outer loop searches over $p(x^n\|y^{n-1})$8. The paper states that this structure is close to what an inverse method would exploit: solving for the multiplier or gain that yields a desired mutual information, SNR, or power level (Delsad, 2023).

5. Generalized optimization frameworks related to IBAA

Several papers abstract the inverse BA pattern into broader projection or mirror-descent frameworks. On mixture families

$p(x^n\|y^{n-1})$9

the generalized iterative minimization algorithm studies

$n$0

through the update

$n$1

For the channel-capacity choice $n$2, the algorithm reduces to classical AB when $n$3. The paper proves monotonic decrease of the objective under a divergence inequality, gives an $n$4-type guarantee, a geometric-rate result under a stronger condition, and explicit error bounds for approximate $n$5-projections (Hayashi, 2023).

The Bregman-divergence-based AB algorithm reformulates generalized BA over a Bregman system generated by a strictly convex potential $n$6. Its core step is

$n$7

equivalently the minimizer of

$n$8

The paper shows monotonicity when $n$9, proves an $1$0-type suboptimality bound under an additional positivity condition, and emphasizes a minimization-free iteration for certain linearly constrained problems. It applies the method to classical and quantum rate-distortion theory (Hayashi, 2024).

For classical-quantum Petz-Rényi capacity, the mirror-descent formulation minimizes

$1$1

with negative entropy mirror map and exponentiated-gradient update

$1$2

The paper establishes global sublinear convergence of the objective values via relative smoothness and local linear convergence in KL divergence on a truncated simplex under tangent-space nondegeneracy and a spectral lower bound in one regime (Lai et al., 15 Jan 2026). Although not an IBAA in name, it has the same entropic multiplicative structure that underlies inverse BA generalizations.

6. Classical-quantum BA identities and the scope of IBAA

In classical-quantum channel capacity, the forward BA-type iteration already contains relations that can be inverted. For a discrete memoryless classical-quantum channel $1$3, the Holevo information is

$1$4

and the BA-type update is

$1$5

equivalently

$1$6

The stopping rule is based on the gap

$1$7

and the fixed-point optimality condition is

$1$8

with equality for active symbols $1$9. The same paper proves monotone convergence, gives the crude complexity bound $P_{Y|X}(y|x)$0, and shows geometric convergence under linear independence of $P_{Y|X}(y|x)$1. It also reports that the empirical convergence is much faster than the proof-based bound (Li et al., 2019).

The binary-input, two-dimensional-output case provides a more explicit inverse bridge. Under a Bloch-sphere parameterization with radii $P_{Y|X}(y|x)$2 and angle $P_{Y|X}(y|x)$3, the paper sets $P_{Y|X}(y|x)$4 as an approximation and obtains a closed-form approximate optimizer

$P_{Y|X}(y|x)$5

with $P_{Y|X}(y|x)$6 when $P_{Y|X}(y|x)$7. The numerical conclusion is that if $P_{Y|X}(y|x)$8, the approximation error is at most $P_{Y|X}(y|x)$9, and the angle $P_{Y|X}(y|x)$00 appears to have surprisingly little influence on the optimal distribution and on the maximum Holevo quantity (Li et al., 2019). A plausible implication is that inverse recovery of state parameters from observed optimizers may be simpler in that regime.

The broader scope of IBAA is therefore heterogeneous. One strand defines IBAA explicitly as distortion recovery from an observed strategy (Vakilipoor et al., 30 Oct 2025). Another uses “inverse” to denote backward causal recursion or reverse projection (Naiss et al., 2010, Toyota, 2019). A third uses inverse BA in a dual or target-constrained sense, recovering the multiplier or distribution associated with a prescribed operating point (Jiao et al., 11 Aug 2025, Trillingsgaard et al., 2013, Delsad, 2023). The literature does not present a single standardized algorithmic object under the name IBAA; rather, it presents a set of closely related constructions that invert standard BA optimality relations in different information-theoretic settings.