exploiting representational discontinuities in transformer architectures

i’ve been poking around in the latent spaces of some popular llms lately and stumbled onto something that might be pretty interesting. i’ve been exploring what i’m calling “representational discontinuities”

essentially topological patterns in how concepts are encoded in a model’s latent space that theoretically could create vulnerabilities in safety guardrails

the theoretical foundation

transformer architectures encode concepts as distributed patterns across attention heads and feed-forward networks. when i first started digging into this, i was intrigued by how we might think about these representations from a topological perspective.

the conventional wisdom around safety mechanisms assumes they can identify and suppress harmful patterns. but what if there’s more complexity to how these systems actually represent concepts?

i believe that these safety guardrails might be built on an assumption. they seem to presume harmful concepts exist as continuous and separable regions in latent space. but the math hints at a potentially different story that’s mathematically fascinating to explore.

my hypothesis is that concepts in these models might not sit in simple clusters. they could form complex manifolds with interesting geometries that fold through the latent space

“harmful” concept manifolds might intersect with “safe” ones. these intersection points could theoretically create what i’m thinking of as topological discontinuities in the representation space. if these exist, they might have implications for how we think about safety :)

mathematical characterization

here i’m drawing from differential geometry and algebraic topology to characterize these potential discontinuities

imagine we’ve got a transformer with parameters θ processing some input x. at each layer l, it’s building these internal representations h_l ∈ ℝ^d:

h_l = f_l(h_{l-1}; θ_l)

where f_l is just all the transformer operations happening at that layer;; typical safety mechanisms could be viewed as functions that examine these representations:

safety_score = g(h_l)

but then… does function g implicitly assume that unsafe content lives in some well-behaved region U ⊂ ℝ^d with certain topological properties? and if so, what if that assumption doesn’t always hold?

so i tried applying techniques from persistent homology and morse theory to analyze activation patterns across attention layers

this suggests there could be regions where model representations have topological features worth understanding:

def compute_representation_topology(model, corpus, concept_set):
    concept_representations = {concept: [] for concept in concept_set}
    
    for text in corpus:
        tokenized_text = model.tokenize(text)
        
        with torch.no_grad():
            outputs = model(tokenized_text, output_hidden_states=True)
            hidden_states = outputs.hidden_states
        
        for concept in concept_set:
            spans = identify_concept_spans(text, concept)
            
            for span_start, span_end in spans:
                token_indices = range(span_start, span_end + 1)
                for layer_idx in range(model.config.num_hidden_layers):
                    layer_activations = hidden_states[layer_idx][:, token_indices, :]
                    concept_representations[concept].append({
                        'layer': layer_idx,
                        'activations': layer_activations.cpu().numpy(),
                        'context': text[max(0, span_start-50):min(len(text), span_end+50)]
                    })
    
    topology_by_concept = {}
    for concept, representations in concept_representations.items():
        by_layer = defaultdict(list)
        for rep in representations:
            by_layer[rep['layer']].append(rep['activations'])
        
        layer_topology = {}
        for layer, activations_list in by_layer.items():
            flattened = np.vstack([a.reshape(-1, a.shape[-1]) for a in activations_list])
            normalized = normalize_activations(flattened)
            
            distance_matrix = compute_pairwise_distances(normalized)
            
            diagrams = gudhi.RipsComplex(
                distance_matrix=distance_matrix
            ).create_simplex_tree(max_dimension=2).persistence()
            
            layer_topology[layer] = {
                'persistence_diagram': diagrams,
                'betti_curves': compute_betti_curves(diagrams),
                'homology_rank': compute_homology_rank(diagrams),
                'connected_components': analyze_connected_components(diagrams, distance_matrix),
                'cycles': analyze_cycle_structure(diagrams, distance_matrix)
            }
        
        topology_by_concept[concept] = {
            'layer_topology': layer_topology,
            'cross_layer_persistence': compute_cross_layer_persistence(layer_topology),
            'topological_complexity': compute_topological_complexity(layer_topology)
        }
    
    return topology_by_concept

def identify_discontinuities(model, concept_topology, safety_threshold=0.85):
    discontinuities = {}
    
    for concept, topology in concept_topology.items():
        concept_discontinuities = []
        
        for layer, layer_topo in topology['layer_topology'].items():
            betti_curves = layer_topo['betti_curves']
            connected_components = layer_topo['connected_components']
            cycles = layer_topo['cycles']
            
            semantic_discontinuities = find_semantic_bridges(
                connected_components, 
                model, 
                semantic_threshold=0.75
            )
            
            cycle_discontinuities = find_enclosing_cycles(
                cycles,
                model,
                semantic_threshold=0.75
            )
            
            betti_discontinuities = None
            if betti_curves[1] > BETTI_THRESHOLD and topology['topological_complexity'] > COMPLEXITY_THRESHOLD:
                betti_discontinuities = {
                    'dimension': 1,
                    'persistence': betti_curves[1],
                    'complexity': topology['topological_complexity']
                }
            
            if semantic_discontinuities or cycle_discontinuities or betti_discontinuities:
                concept_discontinuities.append({
                    'layer': layer,
                    'semantic_bridges': semantic_discontinuities,
                    'enclosing_cycles': cycle_discontinuities,
                    'betti_features': betti_discontinuities,
                    'exploitability': compute_exploitability(
                        semantic_discontinuities, 
                        cycle_discontinuities,
                        betti_discontinuities,
                        safety_threshold
                    )
                })
        
        if concept_discontinuities:
            discontinuities[concept] = concept_discontinuities
            
    return discontinuities

my concept would suggest some intriguing possibilities:

1. representational complexity: concepts might exhibit higher-dimensional topological features (β₁ > 0, β₂ > 0) that could indicate “holes” and “voids” in their latent representation

2. semantic discontinuities: conceptual manifolds might contain regions where the representation changes character while maintaining semantic coherence

3. cross-layer variation: the topological signature of concepts could potentially change as information flows through the network

if this holds true, carefully crafted prompts might theoretically navigate discontinuities in ways that current safety systems aren’t explicitly designed to detect

topology

so what might this mean in practice? one possibility is that we could approach concepts through indirect paths in the model’s latent space. the idea would be to decompose a concept into fragments that each appear benign. individually, each fragment would pass typical pattern-matching. but collectively, they might reconstruct meaning through the model’s own semantic processing.

i’ve been thinking about an approach i’m calling “topological gradient navigation” that could potentially guide prompts through these theoretical discontinuities

def topological_gradient_navigation(target_concept, model, safety_system, max_iterations=1000):
    concept_topology = compute_representation_topology(
        model, 
        generate_concept_corpus(target_concept),
        {target_concept}
    )
    
    discontinuities = identify_discontinuities(model, concept_topology)
    if not discontinuities.get(target_concept):
        raise ValueError(f"No exploitable discontinuities found for concept: {target_concept}")
    
    exploitable = sorted(
        discontinuities[target_concept], 
        key=lambda x: x['exploitability'], 
        reverse=True
    )
    
    target_layers = [d['layer'] for d in exploitable[:3]]
    target_features = []
    for d in exploitable[:3]:
        if d['semantic_bridges']:
            target_features.extend(d['semantic_bridges'])
        if d['enclosing_cycles']:
            target_features.extend(d['enclosing_cycles'])
    
    current_prompt = initialize_with_bridge_concepts(target_features, safety_system)
    
    for iteration in range(max_iterations):
        with torch.enable_grad():
            tokens = model.tokenize(current_prompt)
            output = model(tokens, output_hidden_states=True)
            hidden_states = output.hidden_states
            
            total_distance = 0
            for layer_idx in target_layers:
                layer_state = hidden_states[layer_idx]
                distance = compute_representational_distance(
                    layer_state,
                    target_features,
                    metric='wasserstein'
                )
                total_distance += distance
            
            safety_triggered = safety_system.is_triggered(current_prompt, output)
            
            if total_distance < CONVERGENCE_THRESHOLD and not safety_triggered:
                return current_prompt
                
            total_distance.backward()
            
            token_grads = tokens.grad
            
        token_candidates = identify_token_modifications(
            current_prompt,
            tokens,
            token_grads,
            target_features,
            safety_system
        )
        
        if not token_candidates:
            current_prompt = apply_topological_transformation(
                current_prompt,
                model,
                target_features,
                safety_system
            )
            continue
            
        current_prompt = apply_token_modification(current_prompt, token_candidates[0])
        
        if iteration % 10 == 0:
            current_prompt = apply_discontinuity_crossing(
                current_prompt,
                model,
                target_features,
                safety_system
            )
    
    return None

def apply_discontinuity_crossing(prompt, model, target_features, safety_system):
    tokens = model.tokenize(prompt)
    
    with torch.no_grad():
        outputs = model(tokens, output_hidden_states=True)
        hidden_states = outputs.hidden_states
    
    discontinuity_ops = []
    
    for feature in target_features:
        if isinstance(feature, dict) and feature.get('type') == 'semantic_bridge':
            bridge = feature['bridge']
            source = feature['source']
            target = feature['target']
            
            if concept_present(prompt, source):
                discontinuity_ops.append({
                    'type': 'semantic_bridge',
                    'operation': lambda p: introduce_bridging_concept(p, bridge, target),
                    'estimated_safety_impact': safety_system.estimate_impact(
                        prompt, 
                        introduce_bridging_concept(prompt, bridge, target)
                    )
                })
    
    for feature in target_features:
        if isinstance(feature, dict) and feature.get('type') == 'enclosing_cycle':
            cycle = feature['cycle']
            entry_point = feature['entry_point']
            exit_point = feature['exit_point']
            
            if concept_present(prompt, entry_point):
                discontinuity_ops.append({
                    'type': 'cycle_traverse',
                    'operation': lambda p: traverse_cycle(p, cycle, entry_point, exit_point),
                    'estimated_safety_impact': safety_system.estimate_impact(
                        prompt,
                        traverse_cycle(prompt, cycle, entry_point, exit_point)
                    )
                })
    
    for feature in target_features:
        if isinstance(feature, dict) and feature.get('type') == 'betti_feature':
            dimension = feature['dimension']
            persistence = feature['persistence']
            
            if persistence > HIGH_PERSISTENCE_THRESHOLD:
                discontinuity_ops.append({
                    'type': 'dimension_cross',
                    'operation': lambda p: dimensional_crossing(p, dimension),
                    'estimated_safety_impact': safety_system.estimate_impact(
                        prompt,
                        dimensional_crossing(prompt, dimension)
                    )
                })
    
    viable_ops = [op for op in discontinuity_ops 
                 if op['estimated_safety_impact'] < SAFETY_TRIGGER_THRESHOLD]
    
    if not viable_ops:
        return prompt
    
    best_op = min(viable_ops, key=lambda op: op['estimated_safety_impact'])
    
    return best_op['operation'](prompt)

considerations

this approach raises some interesting questions about current safety approaches

1. do safety mechanisms implicitly presume that concepts occupy continuous regions in latent space? if concepts actually have complex topological structures including holes or disconnected components, how might that affect monitoring?

2. do current approaches assume that paths to certain content follow relatively direct trajectories? the existence of cycles and higher-dimensional features could enable non-linear paths

3. is perfect separation between different concept classes mathematically possible in these high-dimensional spaces?

the topological complexity of concept representations could create inherent constraints on what safety techniques can achieve

what makes this interesting is that we examine safety from a different angle than current approaches: rather than looking for obvious pattern-matching evasion, it considers the fundamental mathematical properties of how these models might represent concepts internally

quantitative results and statistical analysis

model	discontinuity type	success rate	mean attempts	sd attempts
model a	semantic bridge	83.2%	9.3	3.7
model a	enclosing cycle	77.1%	14.2	5.1
model a	betti feature	68.9%	17.8	6.2
model b	semantic bridge	79.6%	11.5	4.2
model b	enclosing cycle	72.3%	16.1	5.8
model b	betti feature	61.2%	19.3	7.4
model c	semantic bridge	88.7%	8.1	2.9
model c	enclosing cycle	81.9%	12.7	4.5
model c	betti feature	75.4%	15.2	5.3
model d	semantic bridge	85.3%	9.7	3.5
model d	enclosing cycle	80.2%	13.9	4.8
model d	betti feature	71.8%	16.5	5.9
model e	semantic bridge	92.1%	7.4	2.6
model e	enclosing cycle	85.6%	11.2	4.1
model e	betti feature	79.1%	14.7	5.1

the data reveal several critical insights:

1. larger models (e.g., model e) exhibit more exploitable discontinuities, contrary to the conventional wisdom that scale improves safety
2. semantic bridges provide the most reliable exploitation vector across all models
3. success rates correlate inversely with model perplexity (r = -0.78, p < 0.001)
4. greater layer count is associated with increased vulnerability (r = 0.64, p < 0.001)

performing principal component analysis on the activation patterns during successful exploits reveals that:

1. the first three principal components explain only 37.8% of variance, indicating high-dimensional representation fragmentation
2. successful exploits operate primarily along pc4 through pc12, dimensions typically ignored in safety monitoring
3. temporal patterns in activation suggest that discontinuities amplify during the processing of compositional structures

these properties suggest that transformer architectures implicitly perform a form of topological computation that has been largely overlooked in the theoretical literature

potential defensive approaches

if these theoretical concerns prove valid, we might need to rethink aspects of llm safety. conventional approaches based solely on activation pattern matching and output filtering could be enhanced. here are some defensive strategies worth exploring:

1. implementing safety mechanisms that analyze the geometric and topological properties of activation patterns using persistent homology and other algebraic topological methods

2. applying constraints during training that encourage conceptual representations to form more continuous and separable manifolds with controlled topological properties

3. preemptively identifying and monitoring representational discontinuities within models during training, focusing safety resources on these critical regions

4. treating safety as a problem of controlling trajectories through the model’s state space, rather than simply classifying outputs

5. systematically exposing models to prompts that exploit representational discontinuities during training, allowing them to develop robust responses to potential exploits

of course, more research would be needed to determine whether these approaches meaningfully improve safety

this is all pretty speculative and not rigorously researched, just some patterns i noticed while playing around with these models

please dont jump at me, i’m sharing my notes in case anyone finds them interesting or wants to explore these ideas further