Domain Expansion: A Latent Space Construction Framework for Multi-Task Learning
Authors: Chi-Yao Huang
Khoa Vo
Aayush Atul Verma
Duo Lu
Yezhou Yang
Venue: ICLR 2026 Poster (2026)
Keywords: representation learning;machine learning for computer vision
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

Abstract: Training a single network with multiple objectives often leads to conflicting gradients that degrade shared representations, forcing them into a compromised state that is suboptimal for any single task—a problem we term latent representation collapse. We introduce Domain Expansion, a framework that prevents these conflicts by restructuring the latent space itself. Our framework uses a novel orthogonal pooling to construct a latent space where each objective is assigned to a mutually orthogonal subspace. We validate our approach on the ShapeNet benchmark, simultaneously training a model for object classification and pose estimation. Our experiments demonstrate that this structure not only prevents collapse but also yields an explicit, interpretable, and compositional latent space where concepts can be directly manipulated.

Published as a conference paper at ICLR 2026
DOMAIN EXPANSION: A LATENT SPACE CONSTRUC- TION FRAMEWORK FOR MULTI-TASK LEARNING
Chi-Yao Huang" Khoa Vo** Aayush Atul Verma Duo Lu², Yezhou Yang'
Arizona State University .uang, ngocbach, averma90, yz ma.yyngaeauu edu [dluer rider. edu}
RRider University
Equal contribution
ABSTRACT
Training single network with multiple objectives often leads to conflicting gra- dients that degrade shared representations, forcing them into compromised state that is suboptimal for any single task-a problem we term latent representation collapse. We introduce Domain Expansion, framework that prevents these con- flicts by restructuring the latent space itself. Our framework uses a novel or- thogonal pooling mechanism to construct latent space where each objective is assigned to a mutually orthogonal subspace. We validate our approach across diverse benchmarks-iinllciiee challenging multi-objective problems combining classification with pose and gaze estimation. Our experiments demonstrate that this structure not only prevents collapse but also yields an explicit, interpretable, and compositional latent space where concepts can be directly manipulated.
ShapeNet, MPIIGaze, and Rotated MNIST-on
INTRODUCTION
Representation learning seeks to map raw data into meaningful latent features in latent space, principle that underpins successes across machine learning, from classification (Chen et al., 2020; Khosla et al., 2020) to large-scale multimodal learning (Radford et al., 2021; Liu et al., 2023b). A common paradigm to train a single, unified network to satisfy multiple learning objectives simul-
(a) Latent representation collapse
(b) Ours (Domain Expansion)
Concept
Concept
All reprasantations satisfy all concepts
Limited representations satisfy all concepts
Concept
Concept 3
Concept
Concept
Figure 1: (a) Latent representation collapse. In standard multi-task learning. competing objec- tives lead latent representation collapse, where the solution spaces for different concepts (colored ellipses) overlap in only a small, compromised region. (b) Domain Expansion. In contrast, our method assigns each concept an orthogonal basis vector in the latent space, preventing inter- ference and creating a structured, interpretable representation where features for each concept are clearly separated.

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Published as a conference paper at ICLR 2026
taneously- for example, performing both classification and regression the same input. However, this approach exposes a fundamental challenge: competing objectives often produce conflicting gra- dients, pulling shared latent features in opposing directions. This interference can degrade model performance, a phenomenon we formalize as latent representation collapse (Fig. (a)).
When collapse occurs, the network carves only small, compromised region of the latent space that partially satisfies all objectives, failing to excel at any. This not only limits predictive accu- racy but also leads to entangled, uninterpretable representations where the underlying factors of variation are obscured. While existing multi-task learning methods attempt to mitigate this by re- weighting task losses or projecting gradients runtime (Yu al., 2020; Sener & Koltun, 2018), these approaches act on the optimization process itself, not the structure of the latent space. The core problem of designing a representation space that is inherently robust interference remains underexplored.
To address this gap, we propose Domain Expansion, a new framework for constructing latent spaces that are multi-objective by design. Instead of mediating gradient conflicts, prevent them struc- turally. The core our framework is orthogonal pooling, a lightweight architectural primitive that constrains features for different objectives to lie in mutually orthogonal subspaces. This enforce- ment of orthogonality ensures that learning one objective cannot interfere with another, allowing the model to learn potent, disentangled features for all tasks simultancously (Fig. 1(b)).
We validate Domain Expansion on a diverse set of benchmarks, including ShapeNet (Chang et al., 2015), MPIIGaze (Zhang al., 2019), and rotated MNIST (Deng, 2012). tackling multi-objective problems that combine classification with various forms of regression like pose and gaze estimation. Across all experiments, our method successfully resolves latent representation collapse. Further- more, we demonstrate that the resulting latent space is not a black box; it possesses an explicit, interpretable structure where distinct objectives are disentangled by design, enabling analysis of their relationships. Our contributions are:
We formalize latent representation collapse, critical failure mode in multi-objective rep- resentation learning.
We introduce Domain Expansion, framework that uses orthogonal pooling construct latent space with mutually orthogonal subspaces, preventing task interference by design.
We demonstrate that our method constructs an explicit and interpretable latent space, where orthogonal axes correspond to distinct concepts, enabling compositional inference and analysis.
2 RELATED WORK
The goal of representation learning is to build latent spaces that capture meaningful factors of vari- ation from raw data. Modern methods often learn such representations by optimizing for multiple, sometimes implicit, objectives. For instance, contrastive learning (Chen et al., 2020; He et al., 2019; Khosla et al., 2020) structures the latent space by pulling similar samples together while pushing dissimilar ones apart. Large multimodal models like CLIP (Radford et al., 2021) learn powerful, transferable features by aligning representations from different modalities. such as images and text. The success of these models hinges on their ability to create unified representation that satisfies these diverse learning signals.
This principle is formalized in multi-task learning (MTL), which aims to improve generalization and efficiency by training single model on multiple tasks simultaneously (Zamir al., 2018). However, central challenge in MTL. is negative transfer (Ruder, 2017), where optimizing for one task degrades the performance of another. This issue often arises from conflicting gradients, where different task-specific losses pull the shared network parameters in opposing directions. This conflict can lead to the phenomenon we term latent representation collapse, where the learned features are a poor compromise that fails to adequately solve any single task.
dominant line of work addresses this challenge by manipulating task gradients during the opti- mization process. These methods aim find less conflicting update vector by altering the gradients from individual tasks. For example, GradNorm (Chen et al., 2017) dynamically adjusts the weights of each task's loss balance their training rates. Other methods focus the geometry of the
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Published as a conference paper at ICLR 2026
Categary Z, Cm
(ours) Guetlevatton
Egi aztimuth
C3 Ggazzimuth
Gerrotation
Gj Rotation
Get Pose
Gy Color
Figure 2: Problem statement. (Left) Real-world inputs contain multiple concepts simultaneously. (Center) Standard multi-objective training leads lalent representation collapse, where concepts interfere and the latent space becomes entangled. (Right) Our Domain Expansion resolves this by assigning each concept to a mutually orthogonal subspace, yielding explicit, interpretable, and compositional latent space.
gradients themselves. PCGrad (Yu et al., 2020) and IMTL (Liu et al., 2021b) identify conflicting gradients and projects them onto the normal plane of others to remove the conflicting components. Building on this, methods like CAGrad (Liu et al., 202 and MGDA (Sener & Koltun, 2018) seek common gradient that represents a Pareto optimal solution minimizes the worst-case loss across tasks.
While effective, these gradient-level methods fundamentally reactive; they resolve conflicts at each training step after they have already occurred. In contrast, our work offers a proactive, representation-centrii solution. Rather than mediating conflicts during optimization, Domain Ex- pansion constructs a latent space with inherent structural properties that prevent interference by design. Through its orthogonal pooling, our framework enforces that different objectives operate in separate, non-in -on-interfering subspaces. This proactive approach eliminates the need for runtime gra- dient manipulation and results in latent space that is more explicit, interpretable, and robust to multi-objective training.
METHOD
Our goal design latent space where a single representation can effectively encode information for multiple, competing objectives without interference. This section details our approach in three parts. First, we formalize the problem of latent representation collapse. Second, we introduce our solution, framework we call Domain Expansion. Finally, we demonstrate how this method yields a structured and compositional latent space with powerful algebraic properties.
3.1 PROBLEM STATEMENT
standard representation learning pipeline involves an encoder, Enc(.), that maps an input from the data space Z to a latent representation f E F RD. For each learning objective m, corresponding decoder, Dec(.), then maps the latent representation f to target concept C, The quality of this mapping is measured by an objective loss L. This can be expressed as:
Ence
Dec C
(evaluated by L)
representation learning, the objective loss L is often defined directly on the latent representations F. The loss function's 's goal is to impose desired structure on the latent space (e.g., clustering or ranking), guided by the supervision from the target concept C.
The challenge arises when this framework is extended to handle set of M objectives simultane- ously. In this scenario, a single, shared representation is subjected to multiple objective losses, ) MM11 each guided by a distinct target concept, {Co,) 1 The total loss becomes a weighted sum applied to these shared representations:
Liotal
Σ MEM
Wm L. (F C.
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(8) Aremorphic Shape
Expansion
Concept: Rectangle
Concegt: G
Concept: Circle
Concept
Figure 3: A single latent vector represents multiple concepts through orthogonal projections, in- spired by anamorphic art. (a) An anamorphic object, such as a cylinder, reveals different primitive concepts (a circle method treats single latent feature as rich object that encodes multiple concepts simultancously. The specific value for each concept is determined by its projection onto corresponding orthogonal axis the latent space.
rectangle) when viewed from orthogonal directions. (b) Analogously,
As we have argued, naively minimizing this sum often leads latent representation collapse, where conflicting structural demands from different losses force the shared representation JF into com- promised state that suboptimal for all objectives (see Fig. 2).
3.2 DOMAIN EXPANSION
To prevent latent collapse in multi-objection learning, we introduce Domain Expansion, a framework that structures the latent space FF into a set of dedicated, orthogonal subspaces. The core idea is to confined to unique subspace, preventing them
ensure that the updates for each objective from interfering with the representations of other objectives (see Fig. 3). Our key insight is that the principal directions of variance in the latent space-its eigenvectors-can be harnessed serve as an orthonormal basis for these dedicated subspaces.
The framework is dynamic process applied at each training epoch, consisting of three steps:
1. Find Principal Axes. First, we estimate the empirical mean and covariance of the latent feature distribution over the current batch entire training set:
= (=) Enc(2)),
Σ = E Enn((2) - #)(Enc(z) 1) T].
(3)
We then perform eigendecomposstion of the covariance matrix to find an orthonormal basis of eigenvectors, [No, "b..",,ii.....
Σ == VVV-1.
(4)
2. Define the Orthogonal Domain. We select the top M eigenvectors (those with the largest eigenvalues) form our conceptual basis, which call the domain, Var {um M). Each eigenvector Om is assigned then to represent a single target concept C... This as- signment ensures that the 1D subspace spanned by um becomes exclusive for all projected features related to that concept. From this, we define the set of orthogonal subspaces and their corresponding projection operators:
F span(um). Proj... = um Vm E M.
(5)
3. Orthogonal Pooling, After defining the orthogonal domain, decompose the latent fea- ture ff into each orthogonal, concept-specific subspace via computing its projection onto each of the domain's s axes:
preoj Projmm (ʃ 1)), Vm M.
(6)
This step, which we term orthogonal pooling, reformulates the learning pipeline as a one- to-many mapping from the shared space the set of orthogonal subspaces, where each
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Abstract Concept Space
Numerical Concept Space
Latent Space
Gj Color
cp = 30cm, color "red""
qe (x 30cm, color "red")
Color
color
Encoding
fi
Labelize
Decoding
Pose
Pose X
Gy: Pose
my:
ef:x 30cm
garolo
Ca:Pose
Figure 4: Concept space vs. latent space: (Left) Real-world inputs possess multiple abstract concepts simultancously, such as color and pose. (Center) For training, define numerical concept space by assigning coordinates to these attributes. (Right) Our model then learns a mapping between this numerical space and its own internal, orthogonal latent space. This process creates an explicit structure where each axis corresponds to a single concept, allowing the model to robustly represent and manipulate the independent attributes of the input.
subspace corresponds to single target concept.
Faan J.Poo
Deco Co
Dees
C1
Orthogonal Pooling
F
P
Decas-1,
CM--
M
The total training loss is then computed as the sum of individual losses on these indepen- dent, projected latent features:
Liotal
Σ Wm f.r (Frrrj C.
(8)
By decomposing the latent space into of dedicated, orthogonal subspaces for each concept, our framework ensures that the resulting loss gradients are inherently decoupled, preventing representa- tion collapse by design.
3.3 PROPERTIES AND OPERATORS OF THE DOMAIN
The orthogonal structure of our method is not merely training aid; it endows the latent space with powerful properties, turning into an interpretable and compositional concept algebra.
To formalize this, we define our terminology in order:
A target concept Cm the set of all possible values for given attribute (e.g., Pose X).
2. A target instantiated concept c is single value from such where et E Cm (c.g.,
x 30 cm).
3. The concept space Cypace is the of all possible combinations of attributes, formally de- fined as the Cartesian Product of the target concept sets (i.c. pose, color, category, etc.):
M M-1-
Cypace = Co x C1 x ... Cas
C..
m=
4. Finally, an instantiated concept c. is a single element from this space, represented as
tuple c = {c "c. c!...., C.....M-M... M-1. During training, each input 2; associated with a specific instantiated concept c.
For each target instantiated concept e of an instantiated concept, we assume a mapping to latent projection frmjj. via the decoder Decm- (The implementation of the decoder, Decm and Dec,.1 detailed Appendix A.
c Decm (J, proj..m
prooj.m Dec, (c
(10)
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Property 1: Orthogonality of Target Concepts. Because each projection subspace Fm is orthog- onal to the others, the target concepts they represent are disentangled in the latent space:
FT
F =
Co 1 C1 T ... Cas-1-
(11)
Property 2: Multi-concept Encoding. A single latent feature Si simultaneously encodes a full instantiated concept Çç. The feature can be decomposed into its orthogonal projections, which are then decoded into their corresponding target instantiated concepts (see Fig. 4):
Si Pooling, (.! ......
Dec {ch (...... - cj.
(12)
Conversely, the full latent feature can be reconstructed from its components:
(C....,,, Dee (sproj.0
p -1 Reconst Si,
(13)
where the reconstruction is defined as:
f # E f.mi.
(14)
MEM
Operator I: Concept-Specific Adjustment (0) and () This operator adjusts an instantiated concept C by applying a change defined by a single target instantiated concept e E Cn- The operation modifies the latent feature fo without affecting any other target concepts. First, we find
the latent vector for the change, freej.m Decm (c The adjusted latent feature is then given by simple vector addition or subtraction:
ff = = 1 proj..
(15)
The full derivation showing the correspondence between the concept and latent spaces as follows:
C
.c....., {c"" @M
........ 1
(16)
Dee
prooj
f
projj
proj. M-1
(17)
Reconst
ft + proj.
(18)
The derivation for the subtraction operator (e") is analogous.
Operator 2: Concept Composition (+) and (0). This operator composes two full instantiated
concepts, Cp and Cg. by operating on their corresponding latent representations, fp and fg. The composition is achieved through simple vector addition or subtraction:
fPq = fp + fq.
(19)
This operation corresponds womponent-wise combination in each orthogonal subspace:
Cp Cq + (1 Φt G),, (cp @t c)...., ( a c9 2
(20)
Dec {[fproj. + fprofju) (spros1 + fq fpros.1) }, [fproj.M -- - fproj.M-1])
(21)
Recoust fp + fg.
(22)
The derivation for the subtraction operator (0) analogous.
EXPERIMENTS
To validate our framework, we formulated and tested three key hypotheses (H):
HI: Does training a single network with multiple objectives lead to latent representation col- lapse as expected?
H2: Does Domain Expansion prevent this collapse and outperform standard multi-task learn- ing baselines?
H3: Does our method create a truly compositional and inferable latent space, as claimed?
Through designing experiments to test these hypotheses, we aimed to systematically evaluate our framework's 's effectiveness and validate its core assumptions. Specifically, we conduct experiments on the ShapeNet dataset (Chang al., 2015), standard benchmark for 3D object classification and pose estimation. We further validate our method on the MPllGaze (Zhang et al., 2019) and Rotated MNIST (Deng, 2012) 2) datasets (see Appendix A.3 for full details).

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Visual input 2
Hungarian Alignment
Latent Construction
Gj: cuteeooryy
Drfo azzmmuck = 10"
fa
Encoder
Var
Ortheconal Fooling
cutegory "chalr"
Vi
Cgi uzimutch
Dety
elevation 45"
V22
Cg elevvation
Figure 5: System overview: An input image is first passed through an encoder to produce latent feature. From the distribution of these features across the dataset, compute the mean (μ) and an orthogonal basis of eigenvectors (Vи)). To ensure consistency during training, this basis sta- bilized across epochs using Hungarian alignment. The orthogonal pooling then projects each latent feature onto these basis vectors. Finally, these projected, non-interfering representations are fed into concept-specific decoders to produce the final outputs.
4.1 EXPERIMENTAL. SETUP
Architecture and Training. We adopt an encoder-decoder architecture here. The encoder, Enc( is a ResNer-50 (He et al., 2015) backbone that produces 2048-dimensional latent feature, f E For each of the M objectives, its corresponding decoder, Decm (+), , a single linear layer. Training proceeds in two stages: first, we train the encoder while dynamically updating the orthogonal basis (VM and μr) at each epoch. To stabilize the basis during early training, we align the eigenvectors between epochs using the Hungarian algorithm (Kubn, 1955) (see Appendix A.2). Second, once the latent space has converged, we freeze the encoder and train the simple linear decoders on the final, fixed representations (see Fig. 5).
Dataset and Concepts. We use the ShapeNet dataset, rendering approximately 30,000 images from 10 models per category using the 3D-R2N2 pipeline (Choy et al., 2016). We define five target cepts: three for regression- azimuth (sampled from [- 2, я /2), elevation, and in in-plane rotation (both sampled from |-x т /T/4) and two for classification (object category and model ID). We denote these concepts as {Caz, Cel, Cro,, Ceat; Caa)
Loss Functions.
To structure the latent space, tation learning losses that operate on the high- high-dimensional latent vectors. Supervised Contrastive (SupCon) (Khosla et al., 2020) imposes clustering constraint using a binary mechanism (positive vs. negative pairs). We adapt the standard SupCon loss by replacing the inner product similarity with L2 distance, which more compatible with our projection-basd based method. Rank-N-Contrast (RNC) (Zha et al., 2023) enforces a more fine-grained ranking mechanism. For both losses, we set the temperature f 2.0. The final loss weights wmm are to 1.0 for RNC and 0.02 SupCon.
weighted combination of two represen-
Based on these components, we define two primary sets of objectives for our experiments, which correspond in order to the concepts {Cau. Cet, Crot. Cear, Ca.. Objective Set 1 uses the RNC loss for all five concepts. Objective Set 2 uses RNC for the first three concepts (regression) and SupCon for the final two (classification).
4.2 EVALUATING OBJECTVBEE PERFORMANCE
To answer our first two research questions (H1 and H2), we evaluate our method against a suite of baselines. We establish simple baseline trained with weighted sum of all losses (Eq. 2). We also compare against three gradient-bare based MTL methods designed to mitigate gradient conflict: Nash- MTL (Navon et al., 2022), FAMO (Liu al., 2023a), and IMTL (Liu et al., 2021b). Finally, evaluate the performance of our full proposed method, Domain Expansion (Ours).
We evaluate the quality of the learned representations for each concept using metries tailored to the task type. For regression concepts (azimuth, elevation, rotation), we measure the Spearman's 's rank correlation (Spearman, 1987). For classification concepts (category, ID), we measure the V-measure

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Table 1: Comprehensive comparison of representation quality, predictive performance, and concept composition. Arrows indicate whether higher (1) or lower (L) values are better.
Representation & Predictive Performance
Concept Comp.
Objective Set
Method
Spearman
t
V-score
MAE
Ace.
Sim.
az
cl
rot
cat
id
az
cl
rot
cat
id
+ and e
baseline
0.41
0.34
0.35 0.16
0.14
0.12
0.09
0.09
0.28
0.37
0.22
FAMO
0.49
0,41
0.42
0.00
0.00
0.12
0.09
0.09
0.19
0.18
0.28
Objective Set
Nash-MTI
0.38
0,41
0,42
0.00
0.00
0.11
0.09
0.09
0.17
0.13
0.28
IMTL
0.31
0.16
0.16
0.39
0.28
0.14
0.11
0.92
0.79
0.14
Ours
0.95
0.87
0.85
0.99
0.91
0.08
0.08
0.09
0.99
0.97
0.95
baseline
0.01
0.01
0.99
0.00
0.77
0.38
0.38
0.99
0.99
0.42
FAMO
0.28
0.23
0.22
0.99
0.00
0.19
0.14
0.13
0.99
0.99
0.28
Objective Set 2
Nash-MTL
0.45
0.39
0.39
0.15
0.00
0.12
0.08
0.09
0.99
0.99
0.35
IMTL
0.39
0.18
0.16
0.99
0.00
0.15
0.11
0.13
0.99 0.99
0.28
Ours
0.95
0.87
0.85
0.98
0.96
0.07
0.08
0.09
0.98
0.94
0.93
(V-score) (Rosenberg & Hirschberg, 2007) to evaluate cluster quality. We also report standard pre- dictive metrics: Mean Absolute Error (MAB") for regression and accuracy (Acc.) for classification.
The results in Table demonstrate the effectiveness of our Domain Expansion framework in prevent- ing latent representation collapse. The baseline model confirms that naive multi-objective training leads to a disorganized latent space, exhibiting poor performance on representation quality metrics like Spearman correlation and V-score. A particularly revealing result visible for Objective Set 2, where several baselines achieve high classification accuracy but have V-score near zero. This discrepancy highlights that they learn a superficial shortcut for the predictive task while their latent space remains collapsed.
In contrast, Domain Expansion significantly outperforms all baselines, achieving superior scores on both the representation metrics and the final predictive tasks. This quantitative success is mirrored in our qualitative visualizations in Fig. 6. Whereas the latent spaces of bascline methods appear entangled and unstructured, the space learned by method is clearly organized, with concepts aligning along their corresponding orthogonal axes.
4.3 PROBING THE COMPOSITIONAL LATENT SPACE
Finally, we designed an experiment to test the composition operator (0 and 2) and verify that our latent space is truly inferable (H3). The goal is if we can transform latent vector fp into target vector fg by applying a conceptual difference defined the concept space.
1. We split the test set into two halves, P and Q. For a pair of samples (zp, cp) and (zq Cq)s our goal synthetically reconstruct fg-
2. The ground-truth target vector is obtained directly from the encoder: fa Enc(zy).
3. We define the conceptual difference between the two samples as ca Cg S Cp-
4. We then create synthetic target concept by applying this difference our source concept:
c = cp @ CA.
5. Using our framework's 's operators, we reconstruct the latent vector corresponding to this synthetic concept: fū Reconst(cç) (Eq. 13).
6. We evaluate the quality of this synthetic reconstruction by computing the average cosine similarity between the reconstructed vectors and the ground-truth vectors over all pairs in the test set Q: ((sp,,.)/oos(f f()]]
As shown in the final column of Table 1, our method's concept composition performance, mea- sured by cosine similarity, is substantially higher than all baselines. This result confirms that our latent space is not a "black box," but rather a meaningful, compositional structure where conceptual operations correspond to simple vector arithmetic, enabling accurate inference and manipulation.
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Baseline Egrrotatoon
Ce aztreur
Cj: elevation
Cjt pategory
Golld
Nash-MTL Cyrooaatonn
Gjuzimmth
Glleration
category
C,id
Domain Expansion (ours)
Cgi aztmmuc
Gieelenation
rotatinio
Cyi category
cild
Figure 6: Qualitative Comparison of Latent Space Structures. We visualize the latent spaces from the baseline, Nash-MTL, and Ours (top to bottom rows) via PCA. Points are colored by their ground-truth concept value, from low (dark purple) to high (bright yellow). Our method (bottom row) demonstrates distinctly more organized structure with clear directional alignment for each concept, unlike the entangled representations produced by the baseline methods.
DISCUSSION AND LIMITATIONS
A key design choice in our framework is leveraging projection-based supervision. Unlike other approaches that might constrain the entire latent vector, method only guides its components along the pre-defined orthogonal axes. It is innately permissive by design: it provides the encoder with significantly more degrees of freedom to learn a rich internal representation, which believe contributes to the performance and compositional properties we observed.
The primary strength of such structured space is its support for high-level conceptual compo- sitions that are difficult for human beings to understand intuitively. For instance, our framework can systematically represent a novel concept like "chair" @ "boat" via the simple latent space op- eration fehair /boat- Our current limitation, therefore, is not in representation but in decoding these abstract concepts. A promising direction for future work to pair our encoder with a genera- tive model, such as LLM diffusion model, to interpret these latent compositions into human- understandable outputs.
CONCLUSION
In this work, we introduced Domain Expansion, a framework that addresses latent representation collapse by constructing latent space with dedicated orthogonal subspaces for each task. Our experiments demonstrated that this design alleviates the risk of task interference and yields ex- plicit and compositional latent space that supports algebraic concept manipulation. This approach suggests a future pathway for learned representation, laying promising foundation for more controllable and interpretable models with applications in areas like algorithmic fairness and controllable multi-modal content generation.
structured bridge between high-level concepts and model's

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7 ACKNOWLEDGMENT
This research is sponsored by NSF, the Partnerships for Innovation grant (#2329780) and TRINA Mothership project. We thank the Research Computing (RC) at Arizona State University (ASU) (Jennewein et al., 2023) and the NSF NAIRR program for their generous support in providing com- puting resources. The views and opinions of the authors expressed herein do not necessarily state or reflect those of the funding agencies and employers.
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Reviews:
[{"review_id": "x4lj2iv0oh", "reviewer": "Reviewer_yLyP", "summary": "The paper introduces Domain Expansion, a novel framework for multi-task learning (MTL) that addresses latent representation collapse categorized as a failure mode where conflicting task gradients lead to entangled, suboptimal shared representations. Instead of modifying gradient updates, the authors propose orthogonal pooling, an architectural mechanism that explicitly decomposes the latent space into mutually orthogonal subspaces, each dedicated to a specific learning objective. This approach aims to prevent interference between tasks structurally, not procedurally. The paper offers a clear three-stage method: (1) compute latent feature covariance, (2) derive an orthonormal basis via eigendecomposition, and (3) project features into task-specific orthogonal subspaces. Each subspace is decoded separately, and training proceeds with independent task losses. Experiments on ShapeNet, MPIIGaze, and Rotated MNIST demonstrate improvements in representation disentanglement and predictive metrics compared to gradient-based MTL baselines (PCGrad, IMTL, Nash-MTL, FAMO). The authors further show that the resulting latent space supports algebraic concept manipulation, such as vector addition and subtraction of concepts.", "questions": "1. **Concept assignment:** How is each eigenvector mapped to a specific task concept? Is there a data-driven matching (e.g., gradient alignment) or manual assignment based on variance ranking?\n\n2. **Computation cost:** What is the training overhead of recomputing eigendecomposition (Eq. 4) per epoch compared to gradient-based MTL baselines?\n\n3. **Dynamic tasks:** Can Domain Expansion adapt if the set of tasks changes mid-training (e.g., new objective added)?\n\n4. **Partial supervision:** Can orthogonal pooling be extended to unsupervised or semi-supervised MTL where some tasks lack labels?\n\n5. **Orthogonality realism:** How sensitive is the method to correlated or redundant tasks? Does strict orthogonality ever hurt performance?\n\n6. **Hungarian alignment:** Why choose Hungarian alignment to track basis order? Would Procrustes or canonical correlation alignment suffice?\n\n7. **Compositional validity:** Beyond cosine similarity, have you tested compositional operators on downstream performance (e.g., predicting unseen concept combinations)?\n\n8. **Numerical stability:** Does covariance estimation introduce instability for large batch or small-batch regimes? Any regularization applied (e.g., shrinkage, clipping)?\n\n9. **Generality:** Can this method be plugged into transformer-based multimodal models (e.g., CLIP, ViT) without modification?\n\n10. **Fair comparison:** Were all baselines tuned to optimal weighting parameters, or did you adopt published defaults?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. **Novel and intuitive framing:** The notion of latent representation collapse as a geometric phenomenon in shared latent spaces is well-motivated and connects neatly to known issues in MTL such as negative transfer and conflicting gradients.\n\n2. **Architectural elegance:** Orthogonal pooling is conceptually simple yet effective. It shifts focus from gradient-level interventions (as in PCGrad or Nash-MTL) to a proactive representation-level solution.\n\n3. **Compositional algebra:** The inclusion of concept-level operations ($\\oplus$, $\\ominus$) adds a unique interpretability dimension to MTL. Section 3.3 formalizes this idea cleanly, showing how orthogonal projections allow structured latent arithmetic.\n\n4. **Methodological clarity:** The training pipeline (Fig. 5) and algorithmic description (Eqs. 3 to 8) are logically structured and mathematically well-grounded.\n\n5. **Empirical experiments:** The method outperforms gradient-based MTL baselines on both representation and predictive metrics (Table 1), with qualitative PCA visualizations (Fig. 6) convincingly showing disentanglement.\n\n6. **Interpretability and compositionality:** Demonstrating vector arithmetic over latent subspaces (Sec. 4.3) is original and compelling; it strengthens the claim that the method yields a structured, interpretable representation.\n\n7. **Presentation quality:** The paper is exceptionally well written and illustrated. Figures 1-4 intuitively connect abstract mathematical ideas to geometric and conceptual metaphors (e.g., the “anamorphic art” analogy in Fig. 3 is particularly effective).", "weaknesses": "# Major Concerns\n\n1. **Empirical scope is narrow.** \nExperiments focus exclusively on relatively small, controlled datasets (ShapeNet, MPIIGaze, Rotated MNIST). These are synthetic or low-dimensional settings. The paper’s claims of scalability and generality (e.g., toward fairness or multimodal learning in Sec. 6) are not yet substantiated.\n\n2. **Unclear stability and computational cost.** \nThe method requires per-epoch covariance estimation and eigendecomposition of large latent spaces (2048-D ResNet-50 features). This step may be $\\mathcal{O}(D^3)$ and unstable for high-dimensional features. The paper should clarify computational cost and numerical stability, especially since Hungarian alignment is used to track basis permutations.\n\n3. **Ambiguity in orthogonality enforcement.**\nThe orthogonal subspaces are defined using the empirical covariance’s eigenvectors. Thus, orthogonality is not learned but imposed externally. This means the decomposition depends heavily on data statistics rather than explicit task semantics. It’s unclear whether orthogonality aligns with task-specific gradients or merely decorrelates features statistically.\n\n4. **Potential circularity in “concept assignment.”**\nThe mapping between eigenvectors (principal components) and tasks (concepts) appears heuristic: the top-M eigenvectors are “assigned” to the M objectives. This lacks theoretical grounding. Why should the top eigenvector correspond to azimuth rather than color, for example? A data- or loss-driven criterion would strengthen this link.\n\n5. **Overstated compositionality claims.**\nThe algebraic operators rely on linear subspace assumptions that may not hold for nonlinear encoders and decoders. Though conceptually elegant, the empirical evidence (Table 1 last column) is limited to cosine similarity metrics. A visual or task-level demonstration (e.g., compositional generation or interpolation) would better support the claim.\n\n6. **Baselines and statistical rigor.**\nResults in Table 1 show large performance gaps (e.g., Spearman 0.95 vs 0.49), which seem unusually high. The paper should include standard deviations, repeated trials, and clarification on hyperparameter tuning fairness. Without this, the reported improvements risk appearing overstated.\n\n7. **Orthogonality and task granularity.**\nThe method presumes one axis per task (Eq. 5), which implicitly assumes tasks are linearly independent. Many real-world MTL setups involve correlated objectives (e.g., segmentation + depth). The framework does not discuss how partially dependent tasks are handled or whether strict orthogonality might hurt shared feature reuse.\n\n8. **Dependence on labeled concept spaces.**\nThe approach assumes explicit supervision for each “concept” (Sec. 3.3), limiting its applicability to unsupervised or weakly labeled MTL. The contrastive + ranking objectives help, but it remains unclear how Domain Expansion would generalize beyond fully annotated multi-concept datasets.\n\n## Minor Concerns\n\n1. **Equation references:** Eq. (8) could clarify whether projection operators $P_m$ are recomputed each epoch or fixed post-training.\n\n2. **Notation clarity:** Use consistent notation between $f_{proj, m}$ and $F_{proj,m}$\n3. **Appendix details:** Appendix A.1 should include pseudocode for the orthogonal pooling operation and basis stabilization via Hungarian alignment.\n4. **Baselines:** Including PCGrad (Yu et al., 2020) explicitly in results tables would give a fairer picture since conceptually it is designed to improve multi-task learning dynamics.\n5. **Discussion:** Section 5’s “chair $\\oplus$ boat” example is charming but anecdotal. Please clarify if such compositions were empirically tested.\n6. **Formatting:** Some tables and figure captions (Table 1, Fig. 6) could benefit from indicating whether higher is better for each metric directly in the caption.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "moD7pt7bmT", "reviewer": "Reviewer_FZR9", "summary": "This paper investigates how to reduce conflicting gradients in multi-objective optimization. Rather than modifying gradients during training as with prior methods, this paper proposes to modify the representation space itself instead by allocating task-specific orthogonal subspaces. Specifically, at each training epoch, i) eigenvectors of the features are computed on the entire training data; ii) the top-m eigenvectors are selected as the task domain and used to construct orthogonal subspaces; iii) each latent feature is then projected onto each orthogonal, concept-specific subspace and the sub-space-specific representations are pooled together; and iv) the projected features are then used to compute the training loss.\n\nThe authors evaluate their approach on ShapeNet, MPIIGaze, and Rotated MNIST where it outperforms baseline and comparison multi-task methods. They also show through quantitative and qualitative analyses that the representation space created by their method is indeed more structured and composable than those of other methods.", "questions": "1. What is the impact of fitting the eigenvectors on the current batch vs larger subsets of the training data vs the entire training dataset?\n    \n2. What is the impact of different choices of the number of eigenvectors M? Is it important for M to be the same as the number of objectives? What happens if M is lower or larger?\n    \n3. How does the approach perform with larger sets of concepts (combining Objective Sets 1 and 2)?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The proposed approach tackles the problem of multi-objective optimization from a different perspective than prior work.\n    \n2. The proposed orthogonal domain is intuitive and well-motivated. The authors illustrate the general problem it should solve and demonstrate through analyses that it is indeed an issue in practice.\n    \n3. The domain expansion approach is principled and the authors describe its operations and properties.\n    \n4. The authors show that their method outperforms baselines both based on predictive performance and in terms of the quality of the representation space.", "weaknesses": "1. Training inefficiency: Eigenvectors need to be calculated and features need to be projected at every epoch. In addition, the Hungarian algorithm needs to be used to align eigenvectors across training epochs. It would be useful to provide some analysis regarding the training times of the proposed method vs the baselines to understand to what extent this is a limitation in practice.\n    \n2. Weak base model: The method is only applied to a fairly old model (ResNet-50). It would be helpful to apply the method to more recent models to see whether it can generalize and is still useful with stronger base models.\n    \n3. Assumption of orthogonality: The method assumes that concepts should be represented based on orthonormal feature spaces. While this may be optimal when concepts and tasks are clearly disentangled as in the experiments, this may reduce sharing of information when objectives are more closely related as is common in real-world applications (e.g., when viewing language modeling, i.e., next-token prediction as multi-task learning, many words rely on shared latent concepts for their prediction). I would appreciate an analysis (can be in a synthetic setting) that provides some insight regarding whether this is an issue in practice.\n    \n4. Narrow evaluation setting: The proposed method is applied to relatively similar datasets containing rotated shapes/digits or gazes. Prior work in this area ([https://arxiv.org/pdf/2001.06782](https://arxiv.org/pdf/2001.06782), [https://arxiv.org/pdf/2010.05874](https://arxiv.org/pdf/2010.05874)) has been evaluated on a broader set of more complex benchmarks including multi-task and multi-label image classification (MultiMNIST, CityScapes, CelebA, multi-task CIFAR-100, NYUv2), multi-task RL, and multi-task NLP datasets. It would be useful to evaluate on more diverse benchmarks to understand the method’s broader applicability and the extent it is applicable to more complex multi-task learning problems.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "iW0vZvb1aI", "reviewer": "Reviewer_CqRr", "summary": "The paper introduces Domain Expansion, a novel framework designed to address the problem of \"latent representation collapse\" in multi-objective learning. The core problem is that when a single network is trained on multiple tasks, conflicting gradients can lead to a compromised latent space that is suboptimal for all tasks. The authors' propose to structure the latent space into a set of mutually orthogonal subspaces, one for each objective. By using an orthogonal pooling mechanism, the method ensures that the learning signals for each task are decoupled, preventing interference by design. The paper demonstrates that this approach not only improves performance on multi-objective benchmarks but also creates an interpretable and compositional latent space where concepts can be manipulated algebraically.", "questions": "* **Q1. Generalizability to Other Architectures:** Could the authors comment on the applicability of Domain Expansion to other encoder architectures, such as Transformers? Does the core assumption that principal eigenvectors correspond to meaningful, disentanglable concepts, hold in those architectures as well?\n\n* **Q2. Performance on Standard Multi-Task Benchmarks:** To better validate the proposed method, it would be insightful to test the model on more complex, standard multi-task benchmarks like NYUv2, CelebA, or Cityscapes. Would the authors anticipate any specific challenges, for instance, with a much larger number of tasks (as in CelebA) or with dense prediction tasks (like semantic segmentation in Cityscapes)?\n\n* **Q3. Transferability of the Learned Representation:** Could a model pre-trained with Domain Expansion be adapted to new tasks or datasets? For example, could a model trained on concepts {azimuth, elevation, category} be effectively fine-tuned on a new dataset that only has labels for {azimuth, category}, or perhaps one where a new concept, {color}, is introduced?\n\n* **Q4. Training Dynamics and Computational Cost:** How many epochs are typically needed for the orthogonal basis to stabilize before the encoder is frozen? What's the time per epoch? What is the computational overhead of the eigendecomposition step per epoch compared to a standard multi-task baseline?\n\n* **Q5. The Value of Representation Quality:** The results for Objective Set 2 compellingly show that baseline methods can achieve high predictive accuracy despite having a collapsed latent representation (i.e., poor V-scores). However, in a practical scenario where a user only cares about the final prediction accuracy on this specific test set, what is the key argument for your method? Does the superior representation quality translate to other critical benefits, such as improved robustness, better generalization to out-of-distribution samples, or greater fairness?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "* **S1. Clarity and Presentation:** The paper is well-written and easy to follow. The authors do a great job of motivating the problem with the clear concept of \"latent representation collapse.\" The flow from the problem statement to the proposed method is logical and intuitive. The figures are highly effective at illustrating the core idea.\n* **2. Originality and Elegance of the Method:** The central idea of harnessing the principal eigenvectors of the latent space's covariance matrix to form an orthogonal basis for different tasks is, to my knowledge, novel. It is a novel approach that addresses the root cause of task interference rather than reactively managing gradient conflicts during optimization, which is the focus of many existing methods.\n* **3. Interpretability and Compositionality:** A significant strength of this work is that it doesn't just aim for better metrics, but for a more structured and meaningful latent space. The demonstration of a \"concept algebra\", where latent vectors can be manipulated through simple arithmetic to compose or adjust concepts, is a contribution towards more interpretable and controllable models.", "weaknesses": "* **W1. Scope of Empirical Evaluation:** The experimental validation, while thorough on the chosen dataset, could be broadened to better establish the generalizability of the method.\n    * **Architectural Diversity**: The experiments are conducted using only a ResNet-50 encoder. It would be beneficial to understand if the method is equally effective with other modern architectures, such as Vision Transformers, which may have different latent space geometries.\n    * **Dataset Diversity**: The primary results are demonstrated on ShapeNet. While MPIIGaze and Rotated MNIST are mentioned in the appendix, a more in-depth analysis on datasets with different characteristics (e.g., more complex scenes, different data modalities) would strengthen the paper's claims.\n    * **Robustness of Results**: The results in Table 1 are reported without standard deviations across multiple runs. Including these would provide greater confidence in the stability and significance of the performance gains.\n* **W2. Clarity of Results in Table 1:** The main results table is dense and could be made more reader-friendly.\n    * **Readability**: A suggestion would be to add a horizontal line to visually separate the baseline methods from the proposed method. Highlighting the best-performing score in each column (e.g., in bold) would also allow for a quicker assessment of the results.\n    * **Caption Details**: The caption could be improved by explicitly defining the acronyms used in the column headers (e.g., az, el, rot, cat, id), as they are not defined in the main text of the experiments section.\n* **W3. Nuanced Performance on Objective Set 2:** The results for Objective Set 2 are interesting and warrant more discussion. While the proposed method clearly excels on representation quality metrics (Spearman, V-score), some baselines achieve slightly better or comparable predictive performance on the downstream tasks (e.g., MAE, Accuracy). This raises the question of the practical trade-offs involved and in which scenarios superior representation quality is most critical.\n* **W4. Reproducibility:** The paper does not mention the release of code. Providing the implementation would be a valuable contribution to the community, allowing others to build upon this work.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "X2BC4ElQFz", "reviewer": "Reviewer_wQsh", "summary": "This paper introduces \"Domain Expansion,\" a novel framework for multi-task learning (MTL) designed to address what the authors term \"latent representation collapse.\" This phenomenon occurs when a single network trained on multiple objectives learns a compromised and suboptimal shared representation due to conflicting gradients. The core idea of Domain Expansion is to prevent this interference structurally by constructing a latent space where each learning objective is assigned to a dedicated, mutually orthogonal subspace. This is achieved through a three-step process at each training epoch: (1) finding the principal axes of the latent feature distribution via eigendecomposition of the covariance matrix, (2) defining an orthogonal domain by assigning the top eigenvectors to different tasks, and (3) using \"orthogonal pooling\" to project the shared latent feature into these concept-specific subspaces before decoding.\n\nThe authors validate their approach on several benchmarks (ShapeNet, MPIIGaze, Rotated MNIST) that combine classification and regression tasks. The experiments show that Domain Expansion not only outperforms standard MTL baselines and gradient-based conflict mitigation methods in both representation quality and predictive performance but also creates a structured and compositional latent space. This structured space allows for algebraic operations on concepts, which the authors demonstrate through concept composition experiments.", "questions": "See Weaknesses", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "* The concept of \"latent representation collapse\" is well-motivated and clearly illustrated in Figure 1. The paper does an excellent job of formalizing this problem and positioning Domain Expansion as a direct solution.\n* A key strength of the proposed method is its ability to create a latent space that is not a \"black box.\" The paper demonstrates that the resulting orthogonal structure enables meaningful, algebraic manipulation of concepts (e.g., concept-specific adjustment and composition), a significant step towards more controllable and interpretable models.", "weaknesses": "*   **Originality and Relation to Prior Work:** The core idea of representing concepts in a latent space may not be entirely new. The paper could be strengthened by discussing its relationship to existing concepts such as superposition in neural networks[1,2], embedding properties in word vectors [3], and the linear representation hypothesis [4].\n*   **Limited Scope of Empirical Validation:** The experiments, while well-executed, are confined to controlled vision datasets (ShapeNet, Rotated MNIST, MPIIGaze). The paper makes broad claims about applicability, but there is no evaluation on other domains like text or multi-modal learning, or on large-scale, complex benchmarks. Furthermore, the lack of an ablation study or resource analysis makes it difficult to assess the scalability and efficiency of the eigendecomposition and Hungarian alignment steps, limiting the claims of general applicability.\n*   **Insufficient Theoretical Grounding for Orthogonal Basis Stability:** The framework relies on a dynamic eigendecomposition performed at each epoch. This raises concerns about the stability of the basis vectors, which could shift and introduce noise into the training process.\n*   **Missing Discussion on Computational and Memory Overheads:** The proposed method introduces a covariance matrix calculation and eigendecomposition step (Equations 3-4) in each training epoch. For high-dimensional latent spaces (D=2048 in this paper), this introduces a significant computational complexity of O(D³), which is not analyzed or compared against the runtime costs of gradient-based MTL methods. This omission may hinder the method's reproducibility and adoption for larger models where this overhead could be prohibitive.\n*   **Minor Ambiguities in Mathematical Formulation:** While the mathematical exposition is generally clear, there are some inconsistencies in notation. For instance, the variable 'F' is used to denote the latent space, a batch of representations, and a single latent vector at different points. Similarly, the formulation of the total loss in Equation (8) seems inconsistent with the definition of the projected subspace in Equation (5), as the loss is computed on projected vectors, not the subspace itself. The notation for the set of objectives 'M' is also used ambiguously. These minor issues could be tightened for improved clarity.\n\n[1] Toy Models of Superposition\n\n[2] Linear algebraic structure of word senses, with applications to polysemy\n\n[3] Linguistic regularities in continuous space word representations\n\n[4] The Linear Representation Hypothesis and the Geometry of Large Language Models", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)